LMFDB - Number field 32.0.845222867573683465013147373404160000000000000000.1

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^32 - 6*x^30 - 64*x^29 + 18*x^28 + 120*x^27 + 884*x^26 + 1360*x^25 + 2453*x^24 - 592*x^23 + 11300*x^22 + 12152*x^21 - 15370*x^20 + 39248*x^19 + 6590*x^18 - 25344*x^17 + 75876*x^16 - 25344*x^15 + 6590*x^14 + 39248*x^13 - 15370*x^12 + 12152*x^11 + 11300*x^10 - 592*x^9 + 2453*x^8 + 1360*x^7 + 884*x^6 + 120*x^5 + 18*x^4 - 64*x^3 - 6*x^2 + 1)

gp: K = bnfinit(y^32 - 6*y^30 - 64*y^29 + 18*y^28 + 120*y^27 + 884*y^26 + 1360*y^25 + 2453*y^24 - 592*y^23 + 11300*y^22 + 12152*y^21 - 15370*y^20 + 39248*y^19 + 6590*y^18 - 25344*y^17 + 75876*y^16 - 25344*y^15 + 6590*y^14 + 39248*y^13 - 15370*y^12 + 12152*y^11 + 11300*y^10 - 592*y^9 + 2453*y^8 + 1360*y^7 + 884*y^6 + 120*y^5 + 18*y^4 - 64*y^3 - 6*y^2 + 1, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^32 - 6*x^30 - 64*x^29 + 18*x^28 + 120*x^27 + 884*x^26 + 1360*x^25 + 2453*x^24 - 592*x^23 + 11300*x^22 + 12152*x^21 - 15370*x^20 + 39248*x^19 + 6590*x^18 - 25344*x^17 + 75876*x^16 - 25344*x^15 + 6590*x^14 + 39248*x^13 - 15370*x^12 + 12152*x^11 + 11300*x^10 - 592*x^9 + 2453*x^8 + 1360*x^7 + 884*x^6 + 120*x^5 + 18*x^4 - 64*x^3 - 6*x^2 + 1);

oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^32 - 6*x^30 - 64*x^29 + 18*x^28 + 120*x^27 + 884*x^26 + 1360*x^25 + 2453*x^24 - 592*x^23 + 11300*x^22 + 12152*x^21 - 15370*x^20 + 39248*x^19 + 6590*x^18 - 25344*x^17 + 75876*x^16 - 25344*x^15 + 6590*x^14 + 39248*x^13 - 15370*x^12 + 12152*x^11 + 11300*x^10 - 592*x^9 + 2453*x^8 + 1360*x^7 + 884*x^6 + 120*x^5 + 18*x^4 - 64*x^3 - 6*x^2 + 1)

$ x^{32} - 6 x^{30} - 64 x^{29} + 18 x^{28} + 120 x^{27} + 884 x^{26} + 1360 x^{25} + 2453 x^{24} + \cdots + 1 $ x^32 - 6*x^30 - 64*x^29 + 18*x^28 + 120*x^27 + 884*x^26 + 1360*x^25 + 2453*x^24 - 592*x^23 + 11300*x^22 + 12152*x^21 - 15370*x^20 + 39248*x^19 + 6590*x^18 - 25344*x^17 + 75876*x^16 - 25344*x^15 + 6590*x^14 + 39248*x^13 - 15370*x^12 + 12152*x^11 + 11300*x^10 - 592*x^9 + 2453*x^8 + 1360*x^7 + 884*x^6 + 120*x^5 + 18*x^4 - 64*x^3 - 6*x^2 + 1

sage: K.defining_polynomial()

gp: K.pol

magma: DefiningPolynomial(K);

oscar: defining_polynomial(K)

Invariants

Degree:	$32$	sage: K.degree() gp: poldegree(K.pol) magma: Degree(K); oscar: degree(K)
Signature:	$[0, 16]$	sage: K.signature() gp: K.sign magma: Signature(K); oscar: signature(K)
Discriminant:	$845222867573683465013147373404160000000000000000$ 845222867573683465013147373404160000000000000000 $\medspace = 2^{64}\cdot 3^{16}\cdot 5^{16}\cdot 17^{8}$	sage: K.disc() gp: K.disc magma: OK := Integers(K); Discriminant(OK); oscar: OK = ring_of_integers(K); discriminant(OK)
Root discriminant:	$31.46$	sage: (K.disc().abs())^(1./K.degree()) gp: abs(K.disc)^(1/poldegree(K.pol)) magma: Abs(Discriminant(OK))^(1/Degree(K)); oscar: (1.0 * dK)^(1/degree(K))
Galois root discriminant:	$2^{2}3^{1/2}5^{1/2}17^{1/2}\approx 63.874877690685246$
Ramified primes:	$2$, $3$, $5$, $17$ 2, 3, 5, 17	sage: K.disc().support() gp: factor(abs(K.disc))[,1]~ magma: PrimeDivisors(Discriminant(OK)); oscar: prime_divisors(discriminant((OK)))
Discriminant root field:	$\Q$
$\card{ \Aut(K/\Q) }$:	$16$	sage: K.automorphisms() magma: Automorphisms(K); oscar: automorphisms(K)
This field is not Galois over $\Q$.
This is a CM field.
Reflex fields:	unavailable$^{32768}$

Integral basis (with respect to field generator $a$)

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2}a^{12}-\frac{1}{2}$, $\frac{1}{2}a^{13}-\frac{1}{2}a$, $\frac{1}{2}a^{14}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{15}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{16}-\frac{1}{2}a^{4}$, $\frac{1}{2}a^{17}-\frac{1}{2}a^{5}$, $\frac{1}{56}a^{18}-\frac{3}{14}a^{17}-\frac{1}{7}a^{16}+\frac{1}{28}a^{15}+\frac{3}{14}a^{13}-\frac{5}{56}a^{12}-\frac{1}{7}a^{10}-\frac{3}{7}a^{9}-\frac{1}{7}a^{8}+\frac{9}{56}a^{6}+\frac{3}{14}a^{5}+\frac{1}{28}a^{3}-\frac{1}{7}a^{2}-\frac{3}{14}a-\frac{13}{56}$, $\frac{1}{56}a^{19}-\frac{3}{14}a^{17}-\frac{5}{28}a^{16}-\frac{1}{14}a^{15}+\frac{3}{14}a^{14}-\frac{1}{56}a^{13}-\frac{1}{14}a^{12}-\frac{1}{7}a^{11}-\frac{1}{7}a^{10}-\frac{2}{7}a^{9}+\frac{2}{7}a^{8}+\frac{9}{56}a^{7}+\frac{1}{7}a^{6}+\frac{1}{14}a^{5}-\frac{13}{28}a^{4}-\frac{3}{14}a^{3}+\frac{1}{14}a^{2}-\frac{17}{56}a+\frac{3}{14}$, $\frac{1}{112}a^{20}-\frac{1}{112}a^{18}+\frac{13}{56}a^{17}-\frac{1}{14}a^{16}-\frac{11}{56}a^{15}-\frac{1}{112}a^{14}-\frac{3}{28}a^{13}-\frac{1}{16}a^{12}-\frac{1}{14}a^{11}-\frac{3}{7}a^{10}-\frac{3}{14}a^{9}+\frac{33}{112}a^{8}-\frac{3}{7}a^{7}+\frac{47}{112}a^{6}-\frac{3}{56}a^{5}-\frac{5}{14}a^{4}+\frac{13}{56}a^{3}-\frac{7}{16}a^{2}+\frac{5}{28}a+\frac{25}{112}$, $\frac{1}{112}a^{21}-\frac{1}{112}a^{19}+\frac{3}{14}a^{17}+\frac{9}{56}a^{16}+\frac{3}{112}a^{15}-\frac{3}{28}a^{14}+\frac{17}{112}a^{13}+\frac{5}{56}a^{12}-\frac{3}{7}a^{11}-\frac{5}{14}a^{10}-\frac{15}{112}a^{9}+\frac{3}{7}a^{8}+\frac{47}{112}a^{7}-\frac{1}{7}a^{6}+\frac{5}{14}a^{5}-\frac{15}{56}a^{4}-\frac{45}{112}a^{3}+\frac{1}{28}a^{2}+\frac{1}{112}a+\frac{1}{56}$, $\frac{1}{112}a^{22}-\frac{1}{112}a^{18}-\frac{1}{28}a^{17}+\frac{19}{112}a^{16}-\frac{13}{56}a^{15}+\frac{1}{7}a^{14}-\frac{5}{56}a^{13}+\frac{9}{112}a^{12}-\frac{3}{7}a^{11}+\frac{17}{112}a^{10}+\frac{5}{14}a^{9}+\frac{3}{7}a^{8}+\frac{3}{7}a^{7}-\frac{17}{112}a^{6}+\frac{3}{28}a^{5}-\frac{29}{112}a^{4}+\frac{19}{56}a^{3}+\frac{2}{7}a^{2}+\frac{15}{56}a-\frac{55}{112}$, $\frac{1}{112}a^{23}-\frac{1}{112}a^{19}+\frac{27}{112}a^{17}-\frac{1}{56}a^{16}+\frac{3}{14}a^{15}-\frac{5}{56}a^{14}+\frac{1}{112}a^{13}-\frac{3}{28}a^{12}+\frac{17}{112}a^{11}+\frac{1}{14}a^{10}-\frac{3}{7}a^{9}+\frac{1}{7}a^{8}-\frac{17}{112}a^{7}+\frac{3}{7}a^{6}-\frac{37}{112}a^{5}-\frac{9}{56}a^{4}+\frac{5}{14}a^{3}-\frac{1}{56}a^{2}-\frac{47}{112}a+\frac{1}{28}$, $\frac{1}{112}a^{24}-\frac{1}{4}a^{15}-\frac{1}{4}a^{12}-\frac{1}{2}a^{9}+\frac{1}{4}a^{3}-\frac{29}{112}$, $\frac{1}{112}a^{25}-\frac{1}{4}a^{16}-\frac{1}{4}a^{13}-\frac{1}{2}a^{10}+\frac{1}{4}a^{4}-\frac{29}{112}a$, $\frac{1}{224}a^{26}-\frac{1}{224}a^{24}-\frac{1}{8}a^{17}-\frac{1}{8}a^{15}+\frac{1}{8}a^{14}-\frac{1}{4}a^{13}-\frac{1}{8}a^{12}-\frac{1}{4}a^{11}-\frac{1}{4}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}+\frac{1}{8}a^{5}+\frac{1}{8}a^{3}+\frac{27}{224}a^{2}+\frac{1}{4}a-\frac{27}{224}$, $\frac{1}{5152}a^{27}-\frac{3}{2576}a^{26}+\frac{19}{5152}a^{25}-\frac{1}{644}a^{24}-\frac{1}{368}a^{23}-\frac{5}{1288}a^{22}-\frac{3}{1288}a^{21}+\frac{9}{2576}a^{20}+\frac{3}{2576}a^{19}+\frac{19}{2576}a^{18}+\frac{321}{2576}a^{17}+\frac{297}{1288}a^{16}-\frac{171}{1288}a^{15}-\frac{37}{368}a^{14}+\frac{431}{2576}a^{13}+\frac{3}{2576}a^{12}-\frac{855}{2576}a^{11}-\frac{311}{1288}a^{10}+\frac{3}{184}a^{9}-\frac{1055}{2576}a^{8}-\frac{773}{2576}a^{7}+\frac{799}{2576}a^{6}+\frac{15}{368}a^{5}+\frac{31}{1288}a^{4}-\frac{813}{5152}a^{3}-\frac{7}{23}a^{2}-\frac{2421}{5152}a+\frac{5}{368}$, $\frac{1}{81091593856}a^{28}-\frac{1111753}{20272898464}a^{27}-\frac{338535}{20272898464}a^{26}-\frac{3747431}{881430368}a^{25}-\frac{80899163}{81091593856}a^{24}+\frac{4339765}{1448064176}a^{23}-\frac{178418691}{40545796928}a^{22}+\frac{29115297}{10136449232}a^{21}-\frac{10236391}{2534112308}a^{20}+\frac{1454717}{194931716}a^{19}+\frac{202137853}{40545796928}a^{18}-\frac{2297265381}{10136449232}a^{17}+\frac{4877298729}{40545796928}a^{16}-\frac{284774923}{1267056154}a^{15}+\frac{28668265}{266748664}a^{14}+\frac{61062193}{5068224616}a^{13}+\frac{10039962313}{40545796928}a^{12}-\frac{4996645883}{10136449232}a^{11}+\frac{1778246291}{5792256704}a^{10}+\frac{5721727}{97465858}a^{9}+\frac{540657589}{2534112308}a^{8}+\frac{2484528465}{10136449232}a^{7}+\frac{4323172117}{40545796928}a^{6}+\frac{704762111}{1448064176}a^{5}+\frac{29368319025}{81091593856}a^{4}-\frac{8270901473}{20272898464}a^{3}+\frac{3186976635}{20272898464}a^{2}-\frac{8894114573}{20272898464}a-\frac{37791327027}{81091593856}$, $\frac{1}{81091593856}a^{29}-\frac{460923}{10136449232}a^{27}-\frac{1321595}{1066994656}a^{26}+\frac{287359505}{81091593856}a^{25}+\frac{5382685}{2896128352}a^{24}+\frac{1839051}{445558208}a^{23}-\frac{167829}{724032088}a^{22}-\frac{1709339}{5068224616}a^{21}+\frac{1793945}{5068224616}a^{20}+\frac{8418663}{1762860736}a^{19}-\frac{4700739}{633528077}a^{18}-\frac{6877769571}{40545796928}a^{17}+\frac{1166494529}{10136449232}a^{16}+\frac{251010219}{5068224616}a^{15}-\frac{426674951}{5068224616}a^{14}-\frac{700302207}{3118907456}a^{13}-\frac{15585905}{110178796}a^{12}+\frac{14764061969}{40545796928}a^{11}+\frac{4548494245}{10136449232}a^{10}+\frac{84710701}{266748664}a^{9}-\frac{4670451529}{10136449232}a^{8}+\frac{14845826313}{40545796928}a^{7}+\frac{1915780785}{5068224616}a^{6}-\frac{2616019239}{81091593856}a^{5}+\frac{8127778}{27544699}a^{4}+\frac{3079714913}{10136449232}a^{3}+\frac{626917365}{2896128352}a^{2}-\frac{16544892471}{81091593856}a+\frac{7335871601}{20272898464}$, $\frac{1}{28\!\cdots\!16}a^{30}+\frac{7557813}{14\!\cdots\!08}a^{29}-\frac{10877665}{28\!\cdots\!16}a^{28}-\frac{30063986275}{408698803178848}a^{27}+\frac{147726305923767}{40\!\cdots\!88}a^{26}-\frac{37\!\cdots\!15}{14\!\cdots\!08}a^{25}+\frac{10\!\cdots\!97}{28\!\cdots\!16}a^{24}-\frac{17\!\cdots\!13}{71\!\cdots\!04}a^{23}+\frac{333003251230917}{20\!\cdots\!44}a^{22}+\frac{37445089707585}{51\!\cdots\!36}a^{21}+\frac{309169187703741}{14\!\cdots\!08}a^{20}+\frac{95300854747025}{54\!\cdots\!08}a^{19}+\frac{22\!\cdots\!51}{35\!\cdots\!52}a^{18}-\frac{15\!\cdots\!71}{71\!\cdots\!04}a^{17}+\frac{51\!\cdots\!93}{46\!\cdots\!68}a^{16}+\frac{18\!\cdots\!47}{89\!\cdots\!88}a^{15}-\frac{31\!\cdots\!01}{46\!\cdots\!68}a^{14}+\frac{309905414443333}{80\!\cdots\!36}a^{13}-\frac{56\!\cdots\!47}{35\!\cdots\!52}a^{12}-\frac{24\!\cdots\!83}{71\!\cdots\!04}a^{11}-\frac{41\!\cdots\!65}{88\!\cdots\!28}a^{10}-\frac{13\!\cdots\!79}{35\!\cdots\!52}a^{9}+\frac{42\!\cdots\!11}{20\!\cdots\!44}a^{8}-\frac{18\!\cdots\!21}{71\!\cdots\!04}a^{7}-\frac{25\!\cdots\!29}{21\!\cdots\!32}a^{6}+\frac{36\!\cdots\!21}{14\!\cdots\!08}a^{5}+\frac{10\!\cdots\!51}{28\!\cdots\!16}a^{4}-\frac{16\!\cdots\!25}{77\!\cdots\!12}a^{3}+\frac{29\!\cdots\!37}{28\!\cdots\!16}a^{2}-\frac{14\!\cdots\!23}{14\!\cdots\!08}a+\frac{83\!\cdots\!83}{28\!\cdots\!16}$, $\frac{1}{13\!\cdots\!52}a^{31}+\frac{1}{13\!\cdots\!52}a^{30}-\frac{213940715}{13\!\cdots\!52}a^{29}+\frac{220682697}{13\!\cdots\!52}a^{28}+\frac{25\!\cdots\!65}{13\!\cdots\!52}a^{27}-\frac{38\!\cdots\!55}{13\!\cdots\!52}a^{26}-\frac{594076212624305}{61\!\cdots\!56}a^{25}-\frac{33\!\cdots\!89}{10\!\cdots\!04}a^{24}+\frac{57\!\cdots\!37}{67\!\cdots\!76}a^{23}+\frac{17\!\cdots\!41}{67\!\cdots\!76}a^{22}+\frac{17\!\cdots\!57}{91\!\cdots\!12}a^{21}-\frac{51\!\cdots\!81}{29\!\cdots\!12}a^{20}-\frac{13\!\cdots\!99}{19\!\cdots\!64}a^{19}+\frac{91\!\cdots\!57}{17\!\cdots\!52}a^{18}+\frac{54\!\cdots\!67}{73\!\cdots\!36}a^{17}+\frac{10\!\cdots\!73}{67\!\cdots\!76}a^{16}+\frac{65\!\cdots\!07}{29\!\cdots\!12}a^{15}+\frac{10\!\cdots\!79}{95\!\cdots\!68}a^{14}-\frac{26\!\cdots\!51}{45\!\cdots\!56}a^{13}+\frac{20\!\cdots\!59}{33\!\cdots\!88}a^{12}+\frac{25\!\cdots\!33}{67\!\cdots\!76}a^{11}-\frac{17\!\cdots\!47}{51\!\cdots\!52}a^{10}-\frac{27\!\cdots\!27}{67\!\cdots\!76}a^{9}+\frac{71\!\cdots\!37}{67\!\cdots\!76}a^{8}+\frac{53\!\cdots\!65}{19\!\cdots\!36}a^{7}-\frac{55\!\cdots\!41}{13\!\cdots\!52}a^{6}-\frac{53\!\cdots\!31}{13\!\cdots\!52}a^{5}+\frac{62\!\cdots\!89}{13\!\cdots\!52}a^{4}-\frac{51\!\cdots\!03}{13\!\cdots\!52}a^{3}+\frac{22\!\cdots\!47}{11\!\cdots\!08}a^{2}-\frac{28\!\cdots\!07}{13\!\cdots\!52}a-\frac{28\!\cdots\!71}{10\!\cdots\!04}$ 1, a, a^2, a^3, a^4, a^5, a^6, a^7, a^8, a^9, a^10, a^11, 1/2*a^12 - 1/2, 1/2*a^13 - 1/2*a, 1/2*a^14 - 1/2*a^2, 1/2*a^15 - 1/2*a^3, 1/2*a^16 - 1/2*a^4, 1/2*a^17 - 1/2*a^5, 1/56*a^18 - 3/14*a^17 - 1/7*a^16 + 1/28*a^15 + 3/14*a^13 - 5/56*a^12 - 1/7*a^10 - 3/7*a^9 - 1/7*a^8 + 9/56*a^6 + 3/14*a^5 + 1/28*a^3 - 1/7*a^2 - 3/14*a - 13/56, 1/56*a^19 - 3/14*a^17 - 5/28*a^16 - 1/14*a^15 + 3/14*a^14 - 1/56*a^13 - 1/14*a^12 - 1/7*a^11 - 1/7*a^10 - 2/7*a^9 + 2/7*a^8 + 9/56*a^7 + 1/7*a^6 + 1/14*a^5 - 13/28*a^4 - 3/14*a^3 + 1/14*a^2 - 17/56*a + 3/14, 1/112*a^20 - 1/112*a^18 + 13/56*a^17 - 1/14*a^16 - 11/56*a^15 - 1/112*a^14 - 3/28*a^13 - 1/16*a^12 - 1/14*a^11 - 3/7*a^10 - 3/14*a^9 + 33/112*a^8 - 3/7*a^7 + 47/112*a^6 - 3/56*a^5 - 5/14*a^4 + 13/56*a^3 - 7/16*a^2 + 5/28*a + 25/112, 1/112*a^21 - 1/112*a^19 + 3/14*a^17 + 9/56*a^16 + 3/112*a^15 - 3/28*a^14 + 17/112*a^13 + 5/56*a^12 - 3/7*a^11 - 5/14*a^10 - 15/112*a^9 + 3/7*a^8 + 47/112*a^7 - 1/7*a^6 + 5/14*a^5 - 15/56*a^4 - 45/112*a^3 + 1/28*a^2 + 1/112*a + 1/56, 1/112*a^22 - 1/112*a^18 - 1/28*a^17 + 19/112*a^16 - 13/56*a^15 + 1/7*a^14 - 5/56*a^13 + 9/112*a^12 - 3/7*a^11 + 17/112*a^10 + 5/14*a^9 + 3/7*a^8 + 3/7*a^7 - 17/112*a^6 + 3/28*a^5 - 29/112*a^4 + 19/56*a^3 + 2/7*a^2 + 15/56*a - 55/112, 1/112*a^23 - 1/112*a^19 + 27/112*a^17 - 1/56*a^16 + 3/14*a^15 - 5/56*a^14 + 1/112*a^13 - 3/28*a^12 + 17/112*a^11 + 1/14*a^10 - 3/7*a^9 + 1/7*a^8 - 17/112*a^7 + 3/7*a^6 - 37/112*a^5 - 9/56*a^4 + 5/14*a^3 - 1/56*a^2 - 47/112*a + 1/28, 1/112*a^24 - 1/4*a^15 - 1/4*a^12 - 1/2*a^9 + 1/4*a^3 - 29/112, 1/112*a^25 - 1/4*a^16 - 1/4*a^13 - 1/2*a^10 + 1/4*a^4 - 29/112*a, 1/224*a^26 - 1/224*a^24 - 1/8*a^17 - 1/8*a^15 + 1/8*a^14 - 1/4*a^13 - 1/8*a^12 - 1/4*a^11 - 1/4*a^9 - 1/2*a^8 - 1/2*a^7 - 1/2*a^6 + 1/8*a^5 + 1/8*a^3 + 27/224*a^2 + 1/4*a - 27/224, 1/5152*a^27 - 3/2576*a^26 + 19/5152*a^25 - 1/644*a^24 - 1/368*a^23 - 5/1288*a^22 - 3/1288*a^21 + 9/2576*a^20 + 3/2576*a^19 + 19/2576*a^18 + 321/2576*a^17 + 297/1288*a^16 - 171/1288*a^15 - 37/368*a^14 + 431/2576*a^13 + 3/2576*a^12 - 855/2576*a^11 - 311/1288*a^10 + 3/184*a^9 - 1055/2576*a^8 - 773/2576*a^7 + 799/2576*a^6 + 15/368*a^5 + 31/1288*a^4 - 813/5152*a^3 - 7/23*a^2 - 2421/5152*a + 5/368, 1/81091593856*a^28 - 1111753/20272898464*a^27 - 338535/20272898464*a^26 - 3747431/881430368*a^25 - 80899163/81091593856*a^24 + 4339765/1448064176*a^23 - 178418691/40545796928*a^22 + 29115297/10136449232*a^21 - 10236391/2534112308*a^20 + 1454717/194931716*a^19 + 202137853/40545796928*a^18 - 2297265381/10136449232*a^17 + 4877298729/40545796928*a^16 - 284774923/1267056154*a^15 + 28668265/266748664*a^14 + 61062193/5068224616*a^13 + 10039962313/40545796928*a^12 - 4996645883/10136449232*a^11 + 1778246291/5792256704*a^10 + 5721727/97465858*a^9 + 540657589/2534112308*a^8 + 2484528465/10136449232*a^7 + 4323172117/40545796928*a^6 + 704762111/1448064176*a^5 + 29368319025/81091593856*a^4 - 8270901473/20272898464*a^3 + 3186976635/20272898464*a^2 - 8894114573/20272898464*a - 37791327027/81091593856, 1/81091593856*a^29 - 460923/10136449232*a^27 - 1321595/1066994656*a^26 + 287359505/81091593856*a^25 + 5382685/2896128352*a^24 + 1839051/445558208*a^23 - 167829/724032088*a^22 - 1709339/5068224616*a^21 + 1793945/5068224616*a^20 + 8418663/1762860736*a^19 - 4700739/633528077*a^18 - 6877769571/40545796928*a^17 + 1166494529/10136449232*a^16 + 251010219/5068224616*a^15 - 426674951/5068224616*a^14 - 700302207/3118907456*a^13 - 15585905/110178796*a^12 + 14764061969/40545796928*a^11 + 4548494245/10136449232*a^10 + 84710701/266748664*a^9 - 4670451529/10136449232*a^8 + 14845826313/40545796928*a^7 + 1915780785/5068224616*a^6 - 2616019239/81091593856*a^5 + 8127778/27544699*a^4 + 3079714913/10136449232*a^3 + 626917365/2896128352*a^2 - 16544892471/81091593856*a + 7335871601/20272898464, 1/2857622031826505216*a^30 + 7557813/1428811015913252608*a^29 - 10877665/2857622031826505216*a^28 - 30063986275/408698803178848*a^27 + 147726305923767/408231718832357888*a^26 - 3753261647483415/1428811015913252608*a^25 + 10552949579894697/2857622031826505216*a^24 - 1746753062829013/714405507956626304*a^23 + 333003251230917/204115859416178944*a^22 + 37445089707585/51028964854044736*a^21 + 309169187703741/1428811015913252608*a^20 + 95300854747025/54954269842817408*a^19 + 2243703231320651/357202753978313152*a^18 - 154562974572713771/714405507956626304*a^17 + 5199026856205393/46090677932685568*a^16 + 1808830912262347/89300688494578288*a^15 - 3124694560078001/46090677932685568*a^14 + 309905414443333/8027028179287936*a^13 - 56506682964527247/357202753978313152*a^12 - 247235924789838483/714405507956626304*a^11 - 4171579863897565/8874602583312128*a^10 - 138908262911127279/357202753978313152*a^9 + 42970402637792811/204115859416178944*a^8 - 186209450362627121/714405507956626304*a^7 - 25106462550943229/219817079371269632*a^6 + 360351479244494121/1428811015913252608*a^5 + 1049766112180212751/2857622031826505216*a^4 - 1663409196355825/7765277260398112*a^3 + 294246258922823537/2857622031826505216*a^2 - 143153873608074723/1428811015913252608*a + 837251644659234383/2857622031826505216, 1/134308235495845745152*a^31 + 1/134308235495845745152*a^30 - 213940715/134308235495845745152*a^29 + 220682697/134308235495845745152*a^28 + 2576441744470865/134308235495845745152*a^27 - 38584408073228855/134308235495845745152*a^26 - 594076212624305/618931960810349056*a^25 - 33901823380287689/10331402730449672704*a^24 + 57676022324789437/67154117747922872576*a^23 + 170706917901480641/67154117747922872576*a^22 + 1760963526035657/919919421204422912*a^21 - 5184893738845981/2919744249909690112*a^20 - 13120292747606999/1975121110233025664*a^19 + 9141844003432157/1767213624945338752*a^18 + 54728212052014767/737957337889262336*a^17 + 10147613037298776873/67154117747922872576*a^16 + 652457099918357207/2919744249909690112*a^15 + 1050533910678523079/9593445392560410368*a^14 - 26886492109286451/459959710602211456*a^13 + 2075000263966150659/33577058873961436288*a^12 + 25560203727769855033/67154117747922872576*a^11 - 1718300521073974547/5165701365224836352*a^10 - 27720020322200391527/67154117747922872576*a^9 + 7150490362531648537/67154117747922872576*a^8 + 5332445055830891965/19186890785120820736*a^7 - 55498703234344568141/134308235495845745152*a^6 - 53125331466494798531/134308235495845745152*a^5 + 62384158746191320889/134308235495845745152*a^4 - 51947126542941615903/134308235495845745152*a^3 + 222781091810137447/1128640634418871808*a^2 - 28093785124115018507/134308235495845745152*a - 2868581203943125971/10331402730449672704

sage: K.integral_basis()

gp: K.zk

magma: IntegralBasis(K);

oscar: basis(OK)

Monogenic:		No
Index:		Not computed
Inessential primes:		$2$

Class group and class number

$C_{4}\times C_{4}$, which has order $16$ (assuming GRH)

sage: K.class_group().invariants()

gp: K.clgp

magma: ClassGroup(K);

oscar: class_group(K)

Unit group

sage: UK = K.unit_group()

magma: UK, fUK := UnitGroup(K);

oscar: UK, fUK = unit_group(OK)

Rank:	$15$	sage: UK.rank() gp: K.fu magma: UnitRank(K); oscar: rank(UK)
Torsion generator:	\( \frac{4689709082965751773}{16788529436980718144} a^{31} + \frac{137330444560895671}{1475914675778524672} a^{30} - \frac{112427100269645412029}{67154117747922872576} a^{29} - \frac{1431214514979260061}{77679719777817088} a^{28} - \frac{15763122802287323847}{16788529436980718144} a^{27} + \frac{4701631034306733264861}{134308235495845745152} a^{26} + \frac{17285813354663541881447}{67154117747922872576} a^{25} + \frac{62136172616210342802389}{134308235495845745152} a^{24} + \frac{27359139605703927634185}{33577058873961436288} a^{23} + \frac{5068993295620531954471}{67154117747922872576} a^{22} + \frac{325445315530141484321}{104276580353917504} a^{21} + \frac{300291824516369256227481}{67154117747922872576} a^{20} - \frac{105790582833901725307985}{33577058873961436288} a^{19} + \frac{162606792401010161338191}{16788529436980718144} a^{18} + \frac{26841638801142444672805}{4796722696280205184} a^{17} - \frac{63155204188786321786795}{9593445392560410368} a^{16} + \frac{23153940802969166122039}{1199180674070051296} a^{15} - \frac{786171987849048597059}{67154117747922872576} a^{14} - \frac{23144614610867763688777}{33577058873961436288} a^{13} + \frac{208568707183896889075829}{16788529436980718144} a^{12} - \frac{4614049092466058863499}{4796722696280205184} a^{11} + \frac{140781568582344069447511}{67154117747922872576} a^{10} + \frac{4670122682691082425955}{987560555116512832} a^{9} + \frac{47048325903156833396441}{67154117747922872576} a^{8} + \frac{3764659978284758030489}{4796722696280205184} a^{7} + \frac{4369682326649553862205}{5839488499819380224} a^{6} + \frac{3410347376610787077089}{9593445392560410368} a^{5} + \frac{2814498244283265128245}{19186890785120820736} a^{4} + \frac{596657558119808319585}{16788529436980718144} a^{3} - \frac{145571178965217497269}{19186890785120820736} a^{2} - \frac{369875046445990575197}{67154117747922872576} a - \frac{35660673875419454845}{134308235495845745152} \) (4689709082965751773)/(16788529436980718144)a^(31) + (137330444560895671)/(1475914675778524672)a^(30) - (112427100269645412029)/(67154117747922872576)a^(29) - (1431214514979260061)/(77679719777817088)a^(28) - (15763122802287323847)/(16788529436980718144)a^(27) + (4701631034306733264861)/(134308235495845745152)a^(26) + (17285813354663541881447)/(67154117747922872576)a^(25) + (62136172616210342802389)/(134308235495845745152)a^(24) + (27359139605703927634185)/(33577058873961436288)a^(23) + (5068993295620531954471)/(67154117747922872576)a^(22) + (325445315530141484321)/(104276580353917504)a^(21) + (300291824516369256227481)/(67154117747922872576)a^(20) - (105790582833901725307985)/(33577058873961436288)a^(19) + (162606792401010161338191)/(16788529436980718144)a^(18) + (26841638801142444672805)/(4796722696280205184)a^(17) - (63155204188786321786795)/(9593445392560410368)a^(16) + (23153940802969166122039)/(1199180674070051296)a^(15) - (786171987849048597059)/(67154117747922872576)a^(14) - (23144614610867763688777)/(33577058873961436288)a^(13) + (208568707183896889075829)/(16788529436980718144)a^(12) - (4614049092466058863499)/(4796722696280205184)a^(11) + (140781568582344069447511)/(67154117747922872576)a^(10) + (4670122682691082425955)/(987560555116512832)a^(9) + (47048325903156833396441)/(67154117747922872576)a^(8) + (3764659978284758030489)/(4796722696280205184)a^(7) + (4369682326649553862205)/(5839488499819380224)a^(6) + (3410347376610787077089)/(9593445392560410368)a^(5) + (2814498244283265128245)/(19186890785120820736)a^(4) + (596657558119808319585)/(16788529436980718144)a^(3) - (145571178965217497269)/(19186890785120820736)a^(2) - (369875046445990575197)/(67154117747922872576)*a - (35660673875419454845)/(134308235495845745152) (order $24$)	sage: UK.torsion_generator() gp: K.tu[2] magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K); oscar: torsion_units_generator(OK)
Fundamental units:	$\frac{10\!\cdots\!99}{67\!\cdots\!76}a^{31}+\frac{10\!\cdots\!39}{25\!\cdots\!76}a^{30}-\frac{70\!\cdots\!19}{83\!\cdots\!72}a^{29}-\frac{18\!\cdots\!51}{19\!\cdots\!36}a^{28}-\frac{44\!\cdots\!93}{11\!\cdots\!84}a^{27}+\frac{15\!\cdots\!25}{10\!\cdots\!04}a^{26}+\frac{46\!\cdots\!01}{33\!\cdots\!88}a^{25}+\frac{33\!\cdots\!37}{13\!\cdots\!52}a^{24}+\frac{11\!\cdots\!35}{23\!\cdots\!92}a^{23}+\frac{66\!\cdots\!51}{67\!\cdots\!76}a^{22}+\frac{47\!\cdots\!69}{25\!\cdots\!76}a^{21}+\frac{66\!\cdots\!35}{29\!\cdots\!12}a^{20}-\frac{37\!\cdots\!13}{33\!\cdots\!88}a^{19}+\frac{50\!\cdots\!91}{83\!\cdots\!72}a^{18}+\frac{59\!\cdots\!25}{32\!\cdots\!72}a^{17}-\frac{84\!\cdots\!81}{67\!\cdots\!76}a^{16}+\frac{36\!\cdots\!75}{33\!\cdots\!88}a^{15}-\frac{12\!\cdots\!87}{67\!\cdots\!76}a^{14}+\frac{16\!\cdots\!91}{37\!\cdots\!92}a^{13}+\frac{20\!\cdots\!65}{41\!\cdots\!36}a^{12}-\frac{14\!\cdots\!27}{83\!\cdots\!72}a^{11}+\frac{33\!\cdots\!29}{95\!\cdots\!68}a^{10}+\frac{48\!\cdots\!25}{33\!\cdots\!88}a^{9}+\frac{35\!\cdots\!13}{30\!\cdots\!28}a^{8}+\frac{29\!\cdots\!81}{29\!\cdots\!12}a^{7}+\frac{43\!\cdots\!15}{13\!\cdots\!52}a^{6}+\frac{27\!\cdots\!69}{72\!\cdots\!28}a^{5}+\frac{18\!\cdots\!83}{13\!\cdots\!52}a^{4}+\frac{47\!\cdots\!43}{67\!\cdots\!76}a^{3}+\frac{10\!\cdots\!33}{13\!\cdots\!52}a^{2}-\frac{11\!\cdots\!47}{83\!\cdots\!72}a-\frac{31\!\cdots\!85}{13\!\cdots\!52}$, $\frac{52\!\cdots\!25}{13\!\cdots\!52}a^{31}-\frac{51\!\cdots\!75}{67\!\cdots\!76}a^{30}-\frac{29\!\cdots\!65}{13\!\cdots\!52}a^{29}-\frac{34\!\cdots\!35}{16\!\cdots\!44}a^{28}+\frac{74\!\cdots\!25}{13\!\cdots\!52}a^{27}+\frac{22\!\cdots\!75}{95\!\cdots\!68}a^{26}+\frac{34\!\cdots\!69}{13\!\cdots\!52}a^{25}-\frac{43\!\cdots\!75}{33\!\cdots\!88}a^{24}+\frac{37\!\cdots\!35}{67\!\cdots\!76}a^{23}-\frac{18\!\cdots\!55}{98\!\cdots\!32}a^{22}+\frac{35\!\cdots\!05}{67\!\cdots\!76}a^{21}-\frac{13\!\cdots\!15}{33\!\cdots\!88}a^{20}-\frac{17\!\cdots\!45}{12\!\cdots\!88}a^{19}+\frac{98\!\cdots\!45}{33\!\cdots\!88}a^{18}-\frac{19\!\cdots\!25}{67\!\cdots\!76}a^{17}-\frac{71\!\cdots\!95}{83\!\cdots\!72}a^{16}+\frac{26\!\cdots\!25}{51\!\cdots\!52}a^{15}-\frac{24\!\cdots\!75}{33\!\cdots\!88}a^{14}+\frac{57\!\cdots\!05}{16\!\cdots\!44}a^{13}+\frac{16\!\cdots\!05}{25\!\cdots\!76}a^{12}-\frac{22\!\cdots\!65}{67\!\cdots\!76}a^{11}+\frac{38\!\cdots\!55}{16\!\cdots\!44}a^{10}-\frac{65\!\cdots\!25}{95\!\cdots\!68}a^{9}-\frac{16\!\cdots\!25}{25\!\cdots\!76}a^{8}+\frac{46\!\cdots\!15}{13\!\cdots\!52}a^{7}-\frac{78\!\cdots\!55}{67\!\cdots\!76}a^{6}-\frac{17\!\cdots\!65}{13\!\cdots\!52}a^{5}-\frac{16\!\cdots\!65}{88\!\cdots\!76}a^{4}+\frac{16\!\cdots\!35}{19\!\cdots\!36}a^{3}+\frac{33\!\cdots\!45}{50\!\cdots\!72}a^{2}+\frac{16\!\cdots\!05}{19\!\cdots\!36}a-\frac{28\!\cdots\!45}{23\!\cdots\!92}$, $\frac{28\!\cdots\!37}{75\!\cdots\!88}a^{31}+\frac{17\!\cdots\!23}{75\!\cdots\!88}a^{30}-\frac{75\!\cdots\!05}{32\!\cdots\!56}a^{29}-\frac{80\!\cdots\!67}{32\!\cdots\!56}a^{28}+\frac{40\!\cdots\!37}{75\!\cdots\!88}a^{27}+\frac{34\!\cdots\!95}{75\!\cdots\!88}a^{26}+\frac{28\!\cdots\!87}{84\!\cdots\!92}a^{25}+\frac{40\!\cdots\!29}{75\!\cdots\!88}a^{24}+\frac{36\!\cdots\!41}{37\!\cdots\!44}a^{23}-\frac{47\!\cdots\!89}{28\!\cdots\!88}a^{22}+\frac{16\!\cdots\!69}{37\!\cdots\!44}a^{21}+\frac{11\!\cdots\!47}{23\!\cdots\!04}a^{20}-\frac{10\!\cdots\!51}{18\!\cdots\!72}a^{19}+\frac{27\!\cdots\!85}{18\!\cdots\!72}a^{18}+\frac{60\!\cdots\!45}{17\!\cdots\!32}a^{17}-\frac{36\!\cdots\!53}{37\!\cdots\!44}a^{16}+\frac{34\!\cdots\!39}{12\!\cdots\!24}a^{15}-\frac{42\!\cdots\!75}{53\!\cdots\!92}a^{14}+\frac{35\!\cdots\!53}{18\!\cdots\!72}a^{13}+\frac{29\!\cdots\!13}{18\!\cdots\!72}a^{12}-\frac{19\!\cdots\!27}{37\!\cdots\!44}a^{11}+\frac{16\!\cdots\!55}{37\!\cdots\!44}a^{10}+\frac{17\!\cdots\!77}{37\!\cdots\!44}a^{9}+\frac{51\!\cdots\!93}{16\!\cdots\!28}a^{8}+\frac{74\!\cdots\!07}{75\!\cdots\!88}a^{7}+\frac{46\!\cdots\!85}{75\!\cdots\!88}a^{6}+\frac{29\!\cdots\!81}{75\!\cdots\!88}a^{5}+\frac{78\!\cdots\!49}{10\!\cdots\!84}a^{4}+\frac{81\!\cdots\!73}{57\!\cdots\!76}a^{3}-\frac{18\!\cdots\!51}{10\!\cdots\!84}a^{2}-\frac{24\!\cdots\!87}{75\!\cdots\!88}a+\frac{10053034696177}{10705016877568}$, $\frac{396174417884787}{29\!\cdots\!24}a^{31}+\frac{10\!\cdots\!41}{14\!\cdots\!72}a^{30}-\frac{35\!\cdots\!17}{67\!\cdots\!76}a^{29}-\frac{27\!\cdots\!41}{54\!\cdots\!16}a^{28}-\frac{47\!\cdots\!39}{10\!\cdots\!84}a^{27}+\frac{17\!\cdots\!31}{13\!\cdots\!52}a^{26}+\frac{65\!\cdots\!27}{67\!\cdots\!76}a^{25}+\frac{87\!\cdots\!95}{13\!\cdots\!52}a^{24}+\frac{34\!\cdots\!17}{33\!\cdots\!88}a^{23}+\frac{11\!\cdots\!73}{67\!\cdots\!76}a^{22}-\frac{35\!\cdots\!07}{16\!\cdots\!44}a^{21}+\frac{54\!\cdots\!27}{67\!\cdots\!76}a^{20}+\frac{29\!\cdots\!51}{33\!\cdots\!88}a^{19}-\frac{16\!\cdots\!17}{16\!\cdots\!44}a^{18}+\frac{92\!\cdots\!27}{33\!\cdots\!88}a^{17}+\frac{33\!\cdots\!61}{67\!\cdots\!76}a^{16}-\frac{42\!\cdots\!33}{26\!\cdots\!21}a^{15}+\frac{35\!\cdots\!59}{67\!\cdots\!76}a^{14}-\frac{61\!\cdots\!65}{37\!\cdots\!92}a^{13}+\frac{96\!\cdots\!97}{16\!\cdots\!44}a^{12}+\frac{87\!\cdots\!91}{33\!\cdots\!88}a^{11}-\frac{60\!\cdots\!47}{67\!\cdots\!76}a^{10}+\frac{84\!\cdots\!07}{98\!\cdots\!32}a^{9}+\frac{50\!\cdots\!59}{67\!\cdots\!76}a^{8}+\frac{13\!\cdots\!53}{33\!\cdots\!88}a^{7}+\frac{91\!\cdots\!15}{58\!\cdots\!24}a^{6}+\frac{59\!\cdots\!55}{67\!\cdots\!76}a^{5}+\frac{98\!\cdots\!65}{13\!\cdots\!52}a^{4}+\frac{43\!\cdots\!33}{52\!\cdots\!42}a^{3}+\frac{16\!\cdots\!79}{13\!\cdots\!52}a^{2}-\frac{32\!\cdots\!17}{67\!\cdots\!76}a-\frac{18\!\cdots\!79}{13\!\cdots\!52}$, $\frac{63\!\cdots\!69}{67\!\cdots\!76}a^{31}+\frac{509368455919691}{19\!\cdots\!52}a^{30}-\frac{12\!\cdots\!83}{23\!\cdots\!92}a^{29}-\frac{64\!\cdots\!21}{10\!\cdots\!04}a^{28}-\frac{35\!\cdots\!49}{21\!\cdots\!96}a^{27}+\frac{18\!\cdots\!27}{19\!\cdots\!36}a^{26}+\frac{29\!\cdots\!73}{33\!\cdots\!88}a^{25}+\frac{90\!\cdots\!27}{58\!\cdots\!24}a^{24}+\frac{50\!\cdots\!87}{16\!\cdots\!44}a^{23}+\frac{19\!\cdots\!53}{35\!\cdots\!04}a^{22}+\frac{38\!\cdots\!15}{33\!\cdots\!88}a^{21}+\frac{95\!\cdots\!97}{67\!\cdots\!76}a^{20}-\frac{25\!\cdots\!71}{33\!\cdots\!88}a^{19}+\frac{78\!\cdots\!67}{20\!\cdots\!68}a^{18}+\frac{95\!\cdots\!11}{83\!\cdots\!72}a^{17}-\frac{58\!\cdots\!13}{67\!\cdots\!76}a^{16}+\frac{22\!\cdots\!37}{33\!\cdots\!88}a^{15}-\frac{84\!\cdots\!83}{67\!\cdots\!76}a^{14}+\frac{51\!\cdots\!35}{19\!\cdots\!68}a^{13}+\frac{25\!\cdots\!11}{83\!\cdots\!72}a^{12}-\frac{68\!\cdots\!35}{41\!\cdots\!36}a^{11}+\frac{14\!\cdots\!39}{67\!\cdots\!76}a^{10}+\frac{29\!\cdots\!91}{33\!\cdots\!88}a^{9}+\frac{15\!\cdots\!43}{21\!\cdots\!96}a^{8}+\frac{41\!\cdots\!85}{67\!\cdots\!76}a^{7}+\frac{26\!\cdots\!83}{13\!\cdots\!52}a^{6}+\frac{86\!\cdots\!13}{37\!\cdots\!03}a^{5}+\frac{11\!\cdots\!71}{13\!\cdots\!52}a^{4}+\frac{29\!\cdots\!17}{67\!\cdots\!76}a^{3}+\frac{68\!\cdots\!33}{13\!\cdots\!52}a^{2}-\frac{13\!\cdots\!37}{16\!\cdots\!44}a-\frac{32\!\cdots\!01}{13\!\cdots\!52}$, $\frac{47\!\cdots\!57}{88\!\cdots\!76}a^{31}+\frac{49\!\cdots\!69}{39\!\cdots\!28}a^{30}-\frac{13\!\cdots\!55}{41\!\cdots\!36}a^{29}-\frac{28\!\cdots\!67}{67\!\cdots\!76}a^{28}-\frac{16\!\cdots\!77}{23\!\cdots\!92}a^{27}+\frac{42\!\cdots\!01}{51\!\cdots\!52}a^{26}+\frac{52\!\cdots\!61}{83\!\cdots\!72}a^{25}+\frac{12\!\cdots\!35}{67\!\cdots\!76}a^{24}+\frac{29\!\cdots\!89}{94\!\cdots\!48}a^{23}+\frac{96\!\cdots\!73}{33\!\cdots\!88}a^{22}+\frac{14\!\cdots\!51}{26\!\cdots\!21}a^{21}+\frac{69\!\cdots\!65}{33\!\cdots\!88}a^{20}+\frac{92\!\cdots\!93}{11\!\cdots\!96}a^{19}+\frac{56\!\cdots\!49}{16\!\cdots\!44}a^{18}+\frac{10\!\cdots\!41}{20\!\cdots\!68}a^{17}-\frac{89\!\cdots\!99}{33\!\cdots\!88}a^{16}+\frac{90\!\cdots\!43}{83\!\cdots\!72}a^{15}+\frac{26\!\cdots\!85}{33\!\cdots\!88}a^{14}-\frac{19\!\cdots\!39}{83\!\cdots\!72}a^{13}+\frac{49\!\cdots\!01}{16\!\cdots\!44}a^{12}+\frac{17\!\cdots\!13}{41\!\cdots\!36}a^{11}-\frac{37\!\cdots\!11}{33\!\cdots\!88}a^{10}+\frac{36\!\cdots\!03}{16\!\cdots\!36}a^{9}+\frac{72\!\cdots\!99}{47\!\cdots\!84}a^{8}+\frac{45\!\cdots\!07}{54\!\cdots\!24}a^{7}+\frac{34\!\cdots\!27}{67\!\cdots\!76}a^{6}+\frac{19\!\cdots\!55}{83\!\cdots\!72}a^{5}+\frac{84\!\cdots\!41}{67\!\cdots\!76}a^{4}+\frac{78\!\cdots\!49}{16\!\cdots\!44}a^{3}-\frac{16\!\cdots\!45}{29\!\cdots\!12}a^{2}-\frac{33\!\cdots\!07}{83\!\cdots\!72}a-\frac{89\!\cdots\!83}{67\!\cdots\!76}$, $\frac{21\!\cdots\!07}{13\!\cdots\!52}a^{31}-\frac{16\!\cdots\!09}{13\!\cdots\!52}a^{30}-\frac{12\!\cdots\!37}{13\!\cdots\!52}a^{29}-\frac{75\!\cdots\!05}{79\!\cdots\!56}a^{28}+\frac{14\!\cdots\!51}{13\!\cdots\!52}a^{27}+\frac{21\!\cdots\!91}{13\!\cdots\!52}a^{26}+\frac{90\!\cdots\!07}{70\!\cdots\!08}a^{25}+\frac{15\!\cdots\!01}{13\!\cdots\!52}a^{24}+\frac{23\!\cdots\!73}{95\!\cdots\!68}a^{23}-\frac{35\!\cdots\!07}{95\!\cdots\!68}a^{22}+\frac{18\!\cdots\!77}{95\!\cdots\!68}a^{21}+\frac{38\!\cdots\!63}{67\!\cdots\!76}a^{20}-\frac{12\!\cdots\!71}{33\!\cdots\!88}a^{19}+\frac{28\!\cdots\!93}{33\!\cdots\!88}a^{18}-\frac{26\!\cdots\!09}{67\!\cdots\!76}a^{17}-\frac{28\!\cdots\!69}{67\!\cdots\!76}a^{16}+\frac{60\!\cdots\!03}{39\!\cdots\!28}a^{15}-\frac{91\!\cdots\!17}{67\!\cdots\!76}a^{14}+\frac{77\!\cdots\!75}{14\!\cdots\!56}a^{13}+\frac{17\!\cdots\!93}{33\!\cdots\!88}a^{12}-\frac{67\!\cdots\!91}{95\!\cdots\!68}a^{11}+\frac{29\!\cdots\!91}{67\!\cdots\!76}a^{10}+\frac{35\!\cdots\!09}{29\!\cdots\!12}a^{9}-\frac{84\!\cdots\!17}{67\!\cdots\!76}a^{8}+\frac{85\!\cdots\!21}{13\!\cdots\!52}a^{7}-\frac{13\!\cdots\!59}{17\!\cdots\!36}a^{6}+\frac{19\!\cdots\!19}{13\!\cdots\!52}a^{5}-\frac{86\!\cdots\!37}{13\!\cdots\!52}a^{4}+\frac{52\!\cdots\!07}{13\!\cdots\!52}a^{3}-\frac{14\!\cdots\!25}{10\!\cdots\!04}a^{2}+\frac{88\!\cdots\!43}{13\!\cdots\!52}a-\frac{40\!\cdots\!93}{13\!\cdots\!52}$, $\frac{35\!\cdots\!37}{67\!\cdots\!76}a^{31}+\frac{13\!\cdots\!21}{13\!\cdots\!52}a^{30}-\frac{10\!\cdots\!55}{33\!\cdots\!88}a^{29}-\frac{11\!\cdots\!05}{32\!\cdots\!56}a^{28}+\frac{21\!\cdots\!89}{67\!\cdots\!76}a^{27}+\frac{46\!\cdots\!15}{70\!\cdots\!08}a^{26}+\frac{40\!\cdots\!75}{83\!\cdots\!72}a^{25}+\frac{84\!\cdots\!41}{10\!\cdots\!04}a^{24}+\frac{12\!\cdots\!73}{83\!\cdots\!72}a^{23}-\frac{37\!\cdots\!37}{67\!\cdots\!76}a^{22}+\frac{20\!\cdots\!07}{33\!\cdots\!88}a^{21}+\frac{51\!\cdots\!13}{67\!\cdots\!76}a^{20}-\frac{23\!\cdots\!25}{33\!\cdots\!88}a^{19}+\frac{82\!\cdots\!89}{41\!\cdots\!36}a^{18}+\frac{17\!\cdots\!27}{23\!\cdots\!92}a^{17}-\frac{12\!\cdots\!59}{95\!\cdots\!68}a^{16}+\frac{12\!\cdots\!49}{33\!\cdots\!88}a^{15}-\frac{39\!\cdots\!51}{67\!\cdots\!76}a^{14}+\frac{57\!\cdots\!49}{47\!\cdots\!84}a^{13}+\frac{18\!\cdots\!65}{83\!\cdots\!72}a^{12}-\frac{72\!\cdots\!01}{16\!\cdots\!44}a^{11}+\frac{34\!\cdots\!71}{67\!\cdots\!76}a^{10}+\frac{35\!\cdots\!81}{47\!\cdots\!84}a^{9}+\frac{56\!\cdots\!17}{67\!\cdots\!76}a^{8}+\frac{13\!\cdots\!91}{95\!\cdots\!68}a^{7}+\frac{14\!\cdots\!55}{13\!\cdots\!52}a^{6}+\frac{21\!\cdots\!39}{33\!\cdots\!88}a^{5}+\frac{33\!\cdots\!77}{19\!\cdots\!36}a^{4}+\frac{26\!\cdots\!09}{67\!\cdots\!76}a^{3}-\frac{42\!\cdots\!25}{19\!\cdots\!36}a^{2}-\frac{22\!\cdots\!87}{33\!\cdots\!88}a-\frac{16\!\cdots\!41}{13\!\cdots\!52}$, $\frac{10\!\cdots\!01}{13\!\cdots\!52}a^{31}+\frac{10\!\cdots\!37}{72\!\cdots\!28}a^{30}-\frac{71\!\cdots\!13}{13\!\cdots\!52}a^{29}-\frac{57\!\cdots\!03}{95\!\cdots\!68}a^{28}-\frac{43\!\cdots\!91}{60\!\cdots\!12}a^{27}+\frac{39\!\cdots\!56}{26\!\cdots\!21}a^{26}+\frac{16\!\cdots\!51}{19\!\cdots\!36}a^{25}+\frac{15\!\cdots\!07}{67\!\cdots\!76}a^{24}+\frac{23\!\cdots\!91}{67\!\cdots\!76}a^{23}+\frac{34\!\cdots\!63}{14\!\cdots\!56}a^{22}+\frac{49\!\cdots\!85}{67\!\cdots\!76}a^{21}+\frac{27\!\cdots\!33}{10\!\cdots\!84}a^{20}-\frac{12\!\cdots\!85}{16\!\cdots\!44}a^{19}+\frac{18\!\cdots\!91}{33\!\cdots\!88}a^{18}+\frac{45\!\cdots\!63}{67\!\cdots\!76}a^{17}-\frac{61\!\cdots\!73}{20\!\cdots\!08}a^{16}+\frac{16\!\cdots\!17}{67\!\cdots\!76}a^{15}+\frac{82\!\cdots\!19}{83\!\cdots\!72}a^{14}-\frac{10\!\cdots\!23}{16\!\cdots\!44}a^{13}+\frac{18\!\cdots\!83}{33\!\cdots\!88}a^{12}+\frac{27\!\cdots\!79}{67\!\cdots\!76}a^{11}-\frac{99\!\cdots\!39}{33\!\cdots\!88}a^{10}+\frac{22\!\cdots\!57}{67\!\cdots\!76}a^{9}+\frac{95\!\cdots\!87}{88\!\cdots\!76}a^{8}-\frac{55\!\cdots\!05}{13\!\cdots\!52}a^{7}+\frac{17\!\cdots\!19}{33\!\cdots\!88}a^{6}+\frac{27\!\cdots\!67}{13\!\cdots\!52}a^{5}+\frac{40\!\cdots\!77}{56\!\cdots\!04}a^{4}-\frac{21\!\cdots\!03}{13\!\cdots\!52}a^{3}+\frac{39\!\cdots\!59}{23\!\cdots\!92}a^{2}-\frac{63\!\cdots\!77}{10\!\cdots\!04}a+\frac{46\!\cdots\!59}{51\!\cdots\!52}$, $\frac{63\!\cdots\!73}{67\!\cdots\!76}a^{31}+\frac{67\!\cdots\!89}{70\!\cdots\!08}a^{30}-\frac{14\!\cdots\!29}{23\!\cdots\!92}a^{29}-\frac{89\!\cdots\!03}{13\!\cdots\!52}a^{28}-\frac{29\!\cdots\!71}{67\!\cdots\!76}a^{27}+\frac{19\!\cdots\!03}{13\!\cdots\!52}a^{26}+\frac{45\!\cdots\!99}{47\!\cdots\!84}a^{25}+\frac{92\!\cdots\!13}{43\!\cdots\!92}a^{24}+\frac{58\!\cdots\!33}{16\!\cdots\!44}a^{23}+\frac{11\!\cdots\!45}{67\!\cdots\!76}a^{22}+\frac{33\!\cdots\!59}{33\!\cdots\!88}a^{21}+\frac{15\!\cdots\!27}{67\!\cdots\!76}a^{20}-\frac{24\!\cdots\!45}{47\!\cdots\!84}a^{19}+\frac{36\!\cdots\!33}{16\!\cdots\!44}a^{18}+\frac{40\!\cdots\!77}{83\!\cdots\!72}a^{17}-\frac{18\!\cdots\!19}{67\!\cdots\!76}a^{16}+\frac{75\!\cdots\!91}{14\!\cdots\!56}a^{15}+\frac{35\!\cdots\!11}{67\!\cdots\!76}a^{14}-\frac{11\!\cdots\!39}{33\!\cdots\!88}a^{13}+\frac{97\!\cdots\!39}{16\!\cdots\!44}a^{12}+\frac{55\!\cdots\!77}{32\!\cdots\!72}a^{11}-\frac{63\!\cdots\!11}{67\!\cdots\!76}a^{10}+\frac{44\!\cdots\!01}{14\!\cdots\!56}a^{9}+\frac{49\!\cdots\!57}{95\!\cdots\!68}a^{8}+\frac{14\!\cdots\!87}{95\!\cdots\!68}a^{7}+\frac{51\!\cdots\!93}{10\!\cdots\!04}a^{6}+\frac{27\!\cdots\!71}{20\!\cdots\!68}a^{5}+\frac{20\!\cdots\!35}{19\!\cdots\!36}a^{4}+\frac{49\!\cdots\!45}{67\!\cdots\!76}a^{3}+\frac{37\!\cdots\!11}{13\!\cdots\!52}a^{2}-\frac{59\!\cdots\!91}{16\!\cdots\!44}a+\frac{10\!\cdots\!21}{13\!\cdots\!52}$, $\frac{37\!\cdots\!59}{67\!\cdots\!76}a^{31}+\frac{54\!\cdots\!27}{25\!\cdots\!76}a^{30}-\frac{11\!\cdots\!11}{33\!\cdots\!88}a^{29}-\frac{48\!\cdots\!49}{13\!\cdots\!52}a^{28}+\frac{18\!\cdots\!53}{21\!\cdots\!96}a^{27}+\frac{12\!\cdots\!43}{19\!\cdots\!36}a^{26}+\frac{83\!\cdots\!77}{16\!\cdots\!44}a^{25}+\frac{10\!\cdots\!97}{13\!\cdots\!52}a^{24}+\frac{12\!\cdots\!23}{83\!\cdots\!72}a^{23}-\frac{20\!\cdots\!87}{95\!\cdots\!68}a^{22}+\frac{21\!\cdots\!21}{33\!\cdots\!88}a^{21}+\frac{47\!\cdots\!05}{67\!\cdots\!76}a^{20}-\frac{26\!\cdots\!91}{33\!\cdots\!88}a^{19}+\frac{37\!\cdots\!67}{16\!\cdots\!44}a^{18}+\frac{28\!\cdots\!59}{72\!\cdots\!28}a^{17}-\frac{11\!\cdots\!11}{95\!\cdots\!68}a^{16}+\frac{14\!\cdots\!87}{33\!\cdots\!88}a^{15}-\frac{91\!\cdots\!91}{67\!\cdots\!76}a^{14}+\frac{20\!\cdots\!77}{33\!\cdots\!88}a^{13}+\frac{36\!\cdots\!93}{16\!\cdots\!44}a^{12}-\frac{12\!\cdots\!77}{16\!\cdots\!44}a^{11}+\frac{55\!\cdots\!11}{67\!\cdots\!76}a^{10}+\frac{20\!\cdots\!85}{33\!\cdots\!88}a^{9}+\frac{69\!\cdots\!83}{21\!\cdots\!96}a^{8}+\frac{13\!\cdots\!07}{67\!\cdots\!76}a^{7}+\frac{11\!\cdots\!59}{13\!\cdots\!52}a^{6}+\frac{21\!\cdots\!23}{33\!\cdots\!88}a^{5}+\frac{23\!\cdots\!87}{13\!\cdots\!52}a^{4}+\frac{52\!\cdots\!09}{95\!\cdots\!68}a^{3}-\frac{28\!\cdots\!03}{13\!\cdots\!52}a^{2}-\frac{50\!\cdots\!51}{33\!\cdots\!88}a-\frac{32\!\cdots\!97}{13\!\cdots\!52}$, $\frac{57\!\cdots\!25}{13\!\cdots\!52}a^{31}+\frac{11\!\cdots\!37}{13\!\cdots\!52}a^{30}-\frac{34\!\cdots\!55}{13\!\cdots\!52}a^{29}-\frac{37\!\cdots\!43}{13\!\cdots\!52}a^{28}+\frac{30\!\cdots\!35}{14\!\cdots\!72}a^{27}+\frac{70\!\cdots\!25}{13\!\cdots\!52}a^{26}+\frac{52\!\cdots\!99}{13\!\cdots\!52}a^{25}+\frac{12\!\cdots\!81}{19\!\cdots\!36}a^{24}+\frac{78\!\cdots\!53}{67\!\cdots\!76}a^{23}-\frac{16\!\cdots\!87}{67\!\cdots\!76}a^{22}+\frac{32\!\cdots\!53}{67\!\cdots\!76}a^{21}+\frac{41\!\cdots\!45}{67\!\cdots\!76}a^{20}-\frac{18\!\cdots\!07}{33\!\cdots\!88}a^{19}+\frac{30\!\cdots\!25}{19\!\cdots\!64}a^{18}+\frac{42\!\cdots\!77}{67\!\cdots\!76}a^{17}-\frac{96\!\cdots\!33}{95\!\cdots\!68}a^{16}+\frac{20\!\cdots\!33}{67\!\cdots\!76}a^{15}-\frac{27\!\cdots\!15}{67\!\cdots\!76}a^{14}+\frac{30\!\cdots\!29}{33\!\cdots\!88}a^{13}+\frac{59\!\cdots\!13}{33\!\cdots\!88}a^{12}-\frac{19\!\cdots\!55}{67\!\cdots\!76}a^{11}+\frac{27\!\cdots\!09}{67\!\cdots\!76}a^{10}+\frac{55\!\cdots\!57}{91\!\cdots\!12}a^{9}+\frac{63\!\cdots\!97}{67\!\cdots\!76}a^{8}+\frac{12\!\cdots\!57}{11\!\cdots\!08}a^{7}+\frac{70\!\cdots\!99}{79\!\cdots\!56}a^{6}+\frac{47\!\cdots\!65}{79\!\cdots\!56}a^{5}+\frac{15\!\cdots\!29}{10\!\cdots\!04}a^{4}+\frac{44\!\cdots\!17}{13\!\cdots\!52}a^{3}-\frac{40\!\cdots\!23}{13\!\cdots\!52}a^{2}-\frac{16\!\cdots\!71}{13\!\cdots\!52}a-\frac{88\!\cdots\!27}{13\!\cdots\!52}$, $\frac{15\!\cdots\!09}{11\!\cdots\!08}a^{31}+\frac{18\!\cdots\!31}{19\!\cdots\!36}a^{30}-\frac{15\!\cdots\!89}{18\!\cdots\!24}a^{29}-\frac{94\!\cdots\!67}{10\!\cdots\!04}a^{28}-\frac{46\!\cdots\!33}{13\!\cdots\!52}a^{27}+\frac{41\!\cdots\!51}{19\!\cdots\!36}a^{26}+\frac{17\!\cdots\!53}{13\!\cdots\!52}a^{25}+\frac{34\!\cdots\!43}{13\!\cdots\!52}a^{24}+\frac{26\!\cdots\!27}{67\!\cdots\!76}a^{23}+\frac{33\!\cdots\!73}{67\!\cdots\!76}a^{22}+\frac{87\!\cdots\!79}{67\!\cdots\!76}a^{21}+\frac{14\!\cdots\!33}{51\!\cdots\!52}a^{20}-\frac{42\!\cdots\!83}{33\!\cdots\!88}a^{19}+\frac{85\!\cdots\!39}{33\!\cdots\!88}a^{18}+\frac{12\!\cdots\!33}{21\!\cdots\!96}a^{17}-\frac{28\!\cdots\!77}{95\!\cdots\!68}a^{16}+\frac{10\!\cdots\!81}{21\!\cdots\!96}a^{15}+\frac{45\!\cdots\!13}{67\!\cdots\!76}a^{14}-\frac{71\!\cdots\!83}{25\!\cdots\!76}a^{13}+\frac{10\!\cdots\!05}{47\!\cdots\!84}a^{12}+\frac{40\!\cdots\!43}{67\!\cdots\!76}a^{11}-\frac{69\!\cdots\!47}{67\!\cdots\!76}a^{10}+\frac{46\!\cdots\!31}{67\!\cdots\!76}a^{9}+\frac{22\!\cdots\!61}{67\!\cdots\!76}a^{8}-\frac{79\!\cdots\!55}{13\!\cdots\!52}a^{7}-\frac{37\!\cdots\!41}{13\!\cdots\!52}a^{6}+\frac{14\!\cdots\!75}{13\!\cdots\!52}a^{5}+\frac{11\!\cdots\!51}{70\!\cdots\!08}a^{4}-\frac{14\!\cdots\!81}{13\!\cdots\!52}a^{3}+\frac{24\!\cdots\!41}{13\!\cdots\!52}a^{2}+\frac{30\!\cdots\!39}{13\!\cdots\!52}a+\frac{10\!\cdots\!97}{18\!\cdots\!04}$, $\frac{12\!\cdots\!63}{33\!\cdots\!88}a^{31}-\frac{35\!\cdots\!23}{16\!\cdots\!44}a^{30}-\frac{53\!\cdots\!65}{23\!\cdots\!92}a^{29}-\frac{79\!\cdots\!21}{33\!\cdots\!88}a^{28}+\frac{26\!\cdots\!83}{33\!\cdots\!88}a^{27}+\frac{73\!\cdots\!77}{16\!\cdots\!44}a^{26}+\frac{18\!\cdots\!63}{57\!\cdots\!62}a^{25}+\frac{16\!\cdots\!47}{33\!\cdots\!88}a^{24}+\frac{10\!\cdots\!15}{11\!\cdots\!64}a^{23}-\frac{42\!\cdots\!53}{16\!\cdots\!44}a^{22}+\frac{41\!\cdots\!03}{98\!\cdots\!32}a^{21}+\frac{21\!\cdots\!25}{49\!\cdots\!16}a^{20}-\frac{98\!\cdots\!75}{16\!\cdots\!44}a^{19}+\frac{25\!\cdots\!49}{16\!\cdots\!44}a^{18}+\frac{10\!\cdots\!21}{70\!\cdots\!88}a^{17}-\frac{15\!\cdots\!77}{16\!\cdots\!44}a^{16}+\frac{49\!\cdots\!11}{16\!\cdots\!44}a^{15}-\frac{73\!\cdots\!75}{64\!\cdots\!44}a^{14}+\frac{58\!\cdots\!13}{16\!\cdots\!44}a^{13}+\frac{25\!\cdots\!97}{16\!\cdots\!44}a^{12}-\frac{87\!\cdots\!59}{11\!\cdots\!96}a^{11}+\frac{94\!\cdots\!75}{16\!\cdots\!44}a^{10}+\frac{97\!\cdots\!29}{23\!\cdots\!92}a^{9}-\frac{33\!\cdots\!75}{49\!\cdots\!16}a^{8}+\frac{16\!\cdots\!43}{11\!\cdots\!28}a^{7}+\frac{89\!\cdots\!45}{20\!\cdots\!68}a^{6}+\frac{53\!\cdots\!05}{16\!\cdots\!44}a^{5}+\frac{29\!\cdots\!51}{33\!\cdots\!88}a^{4}-\frac{96\!\cdots\!25}{25\!\cdots\!76}a^{3}-\frac{10\!\cdots\!55}{16\!\cdots\!44}a^{2}-\frac{33\!\cdots\!35}{16\!\cdots\!44}a+\frac{84\!\cdots\!29}{17\!\cdots\!52}$, $\frac{59\!\cdots\!33}{44\!\cdots\!48}a^{31}-\frac{49\!\cdots\!83}{19\!\cdots\!36}a^{30}-\frac{47\!\cdots\!71}{58\!\cdots\!24}a^{29}-\frac{93\!\cdots\!25}{13\!\cdots\!52}a^{28}+\frac{36\!\cdots\!21}{19\!\cdots\!36}a^{27}+\frac{16\!\cdots\!91}{13\!\cdots\!52}a^{26}+\frac{11\!\cdots\!93}{13\!\cdots\!52}a^{25}-\frac{67\!\cdots\!79}{13\!\cdots\!52}a^{24}-\frac{97\!\cdots\!39}{21\!\cdots\!96}a^{23}-\frac{38\!\cdots\!97}{51\!\cdots\!52}a^{22}+\frac{10\!\cdots\!31}{67\!\cdots\!76}a^{21}-\frac{88\!\cdots\!45}{67\!\cdots\!76}a^{20}-\frac{79\!\cdots\!35}{14\!\cdots\!56}a^{19}+\frac{30\!\cdots\!43}{33\!\cdots\!88}a^{18}-\frac{61\!\cdots\!69}{67\!\cdots\!76}a^{17}-\frac{40\!\cdots\!13}{67\!\cdots\!76}a^{16}+\frac{53\!\cdots\!15}{32\!\cdots\!32}a^{15}-\frac{21\!\cdots\!47}{95\!\cdots\!68}a^{14}+\frac{19\!\cdots\!47}{33\!\cdots\!88}a^{13}+\frac{62\!\cdots\!93}{14\!\cdots\!56}a^{12}-\frac{87\!\cdots\!81}{67\!\cdots\!76}a^{11}+\frac{33\!\cdots\!19}{67\!\cdots\!76}a^{10}-\frac{12\!\cdots\!07}{95\!\cdots\!68}a^{9}-\frac{24\!\cdots\!41}{67\!\cdots\!76}a^{8}+\frac{49\!\cdots\!29}{13\!\cdots\!52}a^{7}-\frac{69\!\cdots\!91}{13\!\cdots\!52}a^{6}-\frac{62\!\cdots\!99}{14\!\cdots\!72}a^{5}-\frac{44\!\cdots\!81}{18\!\cdots\!04}a^{4}-\frac{89\!\cdots\!49}{13\!\cdots\!52}a^{3}-\frac{45\!\cdots\!65}{13\!\cdots\!52}a^{2}+\frac{19\!\cdots\!05}{25\!\cdots\!76}a+\frac{82\!\cdots\!75}{13\!\cdots\!52}$ 10115630303613468499/67154117747922872576a^31 + 10955286308811139/254854336804261376a^30 - 7097749494871534319/8394264718490359072a^29 - 189859000779308695351/19186890785120820736a^28 - 44122616656843393/114013782254537984a^27 + 156037864474214279425/10331402730449672704a^26 + 4672209749539424081601/33577058873961436288a^25 + 33544856023756934694337/134308235495845745152a^24 + 1150818225322384177535/2398361348140102592a^23 + 6621236039032137512651/67154117747922872576a^22 + 4716290987554550034369/2582850682612418176a^21 + 6695799013182615228735/2919744249909690112a^20 - 37943758766504169928313/33577058873961436288a^19 + 50329182181694230630891/8394264718490359072a^18 + 596661939155444835725/322856335326552272a^17 - 84721314070516893525281/67154117747922872576a^16 + 363373349492095772564275/33577058873961436288a^15 - 129879119369970803980387/67154117747922872576a^14 + 1616890361916249496991/377270324426532992a^13 + 20575865915639761862665/4197132359245179536a^12 - 1407226887684003917927/8394264718490359072a^11 + 33304517895382376400829/9593445392560410368a^10 + 48320843743809771335025/33577058873961436288a^9 + 359862580911360355913/309465980405174528a^8 + 2981751610244084671081/2919744249909690112a^7 + 43549926250140913226115/134308235495845745152a^6 + 274887587197085005469/729936062477422528a^5 + 18603632892036024803183/134308235495845745152a^4 + 4709058258287294615043/67154117747922872576a^3 + 1098570249339500988333/134308235495845745152a^2 - 11197323945355880947/8394264718490359072a - 313571325731510717985/134308235495845745152, 5289040508646213225/134308235495845745152a^31 - 5184028121007995275/67154117747922872576a^30 - 29608291856446849865/134308235495845745152a^29 - 34524333423473657535/16788529436980718144a^28 + 746331462436254908025/134308235495845745152a^27 + 22257495712946207575/9593445392560410368a^26 + 3461579496477284585569/134308235495845745152a^25 - 434406713485320432875/33577058873961436288a^24 + 379158327078868167435/67154117747922872576a^23 - 187858571321701236155/987560555116512832a^22 + 35783566374276444619605/67154117747922872576a^21 - 13324142850688464372315/33577058873961436288a^20 - 1752159811812428112345/1291425341306209088a^19 + 98527890391582227947145/33577058873961436288a^18 - 199984012735313370658825/67154117747922872576a^17 - 7166045948004864024295/8394264718490359072a^16 + 26136437166822905566925/5165701365224836352a^15 - 240990016671253937175275/33577058873961436288a^14 + 57943933713980521506905/16788529436980718144a^13 + 1696466819275000178005/2582850682612418176a^12 - 229141615055489306070165/67154117747922872576a^11 + 38751438711071189091055/16788529436980718144a^10 - 6584385913864142107725/9593445392560410368a^9 - 1689913370901992272125/2582850682612418176a^8 + 46692322923547328940415/134308235495845745152a^7 - 7876035856620144643855/67154117747922872576a^6 - 1719787267924375377465/134308235495845745152a^5 - 16274301085059084365/883606812472669376a^4 + 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1876451828280215863/5765291702259862a^25 + 16393212345675024978247/33577058873961436288a^24 + 102551113527216920915/114989927650552864a^23 - 4206758315476042244353/16788529436980718144a^22 + 4193526597426308389203/987560555116512832a^21 + 2119619385848774546825/493780277558256416a^20 - 98871586807681252740275/16788529436980718144a^19 + 254624740033858706191349/16788529436980718144a^18 + 109071683497775067221/70540039651179488a^17 - 158202158842095911345277/16788529436980718144a^16 + 492949362012334524153011/16788529436980718144a^15 - 7398055630702254218075/645712670653104544a^14 + 58147357739364504043313/16788529436980718144a^13 + 255367188738298177929097/16788529436980718144a^12 - 8704240785849789439659/1199180674070051296a^11 + 94590718517947631119675/16788529436980718144a^10 + 9770652744406807881729/2398361348140102592a^9 - 337920243222398061375/493780277558256416a^8 + 16103459178021185243/11902537707891328a^7 + 896626472799870301745/2098566179622589768a^6 + 5307249779375211696505/16788529436980718144a^5 + 2982390572851428211451/33577058873961436288a^4 - 9637855042034936825/2582850682612418176a^3 - 101918974900879492955/16788529436980718144a^2 - 33728189955233097235/16788529436980718144a + 844076693480187729/1767213624945338752, 59585312878670533/449191423063029248a^31 - 4974484531619672683/19186890785120820736a^30 - 4764899942767196571/5839488499819380224a^29 - 931269385373661508625/134308235495845745152a^28 + 366311088118544791121/19186890785120820736a^27 + 1683425560632120371691/134308235495845745152a^26 + 11528155196530127858193/134308235495845745152a^25 - 6794062506904904304779/134308235495845745152a^24 - 97572343157032483739/2166261862836221696a^23 - 3837136002951083577597/5165701365224836352a^22 + 107115017315233048587831/67154117747922872576a^21 - 88572070801167500866545/67154117747922872576a^20 - 7942222074645109175135/1459872124954845056a^19 + 300616965279930618045743/33577058873961436288a^18 - 610751100249106700030669/67154117747922872576a^17 - 402090683973128965950113/67154117747922872576a^16 + 534190607311552614515/32085101647359232a^15 - 219424890814357884032747/9593445392560410368a^14 + 194919571652593059278347/33577058873961436288a^13 + 6243108058074578525293/1459872124954845056a^12 - 875930617929254692601681/67154117747922872576a^11 + 332919658291847359455519/67154117747922872576a^10 - 12724401841212687379607/9593445392560410368a^9 - 245866484130709948405541/67154117747922872576a^8 + 49614090553688773560829/134308235495845745152a^7 - 69935538292328064578991/134308235495845745152a^6 - 623264989979645706799/1475914675778524672a^5 - 44172828323963694681/188370596768367104a^4 - 8963989301880896677049/134308235495845745152a^3 - 4515851576704796012365/134308235495845745152a^2 + 1906151139750041805/254854336804261376*a + 82733429493683238575/134308235495845745152 (assuming GRH)	sage: UK.fundamental_units() gp: K.fu magma: [K\|fUK(g): g in Generators(UK)]; oscar: [K(fUK(a)) for a in gens(UK)]
Regulator:	$ 124577213214.25456 $ (assuming GRH)	sage: K.regulator() gp: K.reg magma: Regulator(K); oscar: regulator(K)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{16}\cdot 124577213214.25456 \cdot 16}{24\cdot\sqrt{845222867573683465013147373404160000000000000000}}\cr\approx \mathstrut & 0.533015240485038 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula

x = polygen(QQ); K.<a> = NumberField(x^32 - 6*x^30 - 64*x^29 + 18*x^28 + 120*x^27 + 884*x^26 + 1360*x^25 + 2453*x^24 - 592*x^23 + 11300*x^22 + 12152*x^21 - 15370*x^20 + 39248*x^19 + 6590*x^18 - 25344*x^17 + 75876*x^16 - 25344*x^15 + 6590*x^14 + 39248*x^13 - 15370*x^12 + 12152*x^11 + 11300*x^10 - 592*x^9 + 2453*x^8 + 1360*x^7 + 884*x^6 + 120*x^5 + 18*x^4 - 64*x^3 - 6*x^2 + 1)

DK = K.disc(); r1,r2 = K.signature(); RK = K.regulator(); RR = RK.parent()

hK = K.class_number(); wK = K.unit_group().torsion_generator().order();

2^r1 * (2*RR(pi))^r2 * RK * hK / (wK * RR(sqrt(abs(DK))))

# self-contained Pari/GP code snippet to compute the analytic class number formula

K = bnfinit(x^32 - 6*x^30 - 64*x^29 + 18*x^28 + 120*x^27 + 884*x^26 + 1360*x^25 + 2453*x^24 - 592*x^23 + 11300*x^22 + 12152*x^21 - 15370*x^20 + 39248*x^19 + 6590*x^18 - 25344*x^17 + 75876*x^16 - 25344*x^15 + 6590*x^14 + 39248*x^13 - 15370*x^12 + 12152*x^11 + 11300*x^10 - 592*x^9 + 2453*x^8 + 1360*x^7 + 884*x^6 + 120*x^5 + 18*x^4 - 64*x^3 - 6*x^2 + 1, 1);

[polcoeff (lfunrootres (lfuncreate (K))[1][1][2], -1), 2^K.r1 * (2*Pi)^K.r2 * K.reg * K.no / (K.tu[1] * sqrt (abs (K.disc)))]

/* self-contained Magma code snippet to compute the analytic class number formula */

Qx<x> := PolynomialRing(QQ); K<a> := NumberField(x^32 - 6*x^30 - 64*x^29 + 18*x^28 + 120*x^27 + 884*x^26 + 1360*x^25 + 2453*x^24 - 592*x^23 + 11300*x^22 + 12152*x^21 - 15370*x^20 + 39248*x^19 + 6590*x^18 - 25344*x^17 + 75876*x^16 - 25344*x^15 + 6590*x^14 + 39248*x^13 - 15370*x^12 + 12152*x^11 + 11300*x^10 - 592*x^9 + 2453*x^8 + 1360*x^7 + 884*x^6 + 120*x^5 + 18*x^4 - 64*x^3 - 6*x^2 + 1);

OK := Integers(K); DK := Discriminant(OK);

UK, fUK := UnitGroup(OK); clK, fclK := ClassGroup(OK);

r1,r2 := Signature(K); RK := Regulator(K); RR := Parent(RK);

hK := #clK; wK := #TorsionSubgroup(UK);

2^r1 * (2*Pi(RR))^r2 * RK * hK / (wK * Sqrt(RR!Abs(DK)));

# self-contained Oscar code snippet to compute the analytic class number formula

Qx, x = PolynomialRing(QQ); K, a = NumberField(x^32 - 6*x^30 - 64*x^29 + 18*x^28 + 120*x^27 + 884*x^26 + 1360*x^25 + 2453*x^24 - 592*x^23 + 11300*x^22 + 12152*x^21 - 15370*x^20 + 39248*x^19 + 6590*x^18 - 25344*x^17 + 75876*x^16 - 25344*x^15 + 6590*x^14 + 39248*x^13 - 15370*x^12 + 12152*x^11 + 11300*x^10 - 592*x^9 + 2453*x^8 + 1360*x^7 + 884*x^6 + 120*x^5 + 18*x^4 - 64*x^3 - 6*x^2 + 1);

OK = ring_of_integers(K); DK = discriminant(OK);

UK, fUK = unit_group(OK); clK, fclK = class_group(OK);

r1,r2 = signature(K); RK = regulator(K); RR = parent(RK);

hK = order(clK); wK = torsion_units_order(K);

2^r1 * (2*pi)^r2 * RK * hK / (wK * sqrt(RR(abs(DK))))

Galois group

$D_4\times C_2^3$ (as 32T273):

sage: K.galois_group(type='pari')

gp: polgalois(K.pol)

magma: G = GaloisGroup(K);

oscar: G, Gtx = galois_group(K); G, transitive_group_identification(G)

A solvable group of order 64

The 40 conjugacy class representatives for $D_4\times C_2^3$

Character table for $D_4\times C_2^3$ is not computed

Intermediate fields

$\Q(\sqrt{2}) $, $\Q(\sqrt{10}) $, $\Q(\sqrt{-10}) $, $\Q(\sqrt{-2}) $, $\Q(\sqrt{15}) $, $\Q(\sqrt{3}) $, $\Q(\sqrt{-3}) $, $\Q(\sqrt{-15}) $, $\Q(\sqrt{5}) $, $\Q(\sqrt{-5}) $, $\Q(\sqrt{30}) $, $\Q(\sqrt{-30}) $, $\Q(\sqrt{-6}) $, $\Q(\sqrt{6}) $, $\Q(\sqrt{-1}) $, 4.0.979200.3, 4.0.39168.3, 4.4.9792.1, 4.4.244800.1, 4.0.1088.2, 4.0.27200.2, 4.4.108800.1, 4.4.4352.1, $\Q(i, \sqrt{5})$, $\Q(\sqrt{5}, \sqrt{6})$, $\Q(\sqrt{5}, \sqrt{-6})$, $\Q(\sqrt{-5}, \sqrt{-6})$, $\Q(\sqrt{-5}, \sqrt{6})$, $\Q(i, \sqrt{30})$, $\Q(i, \sqrt{6})$, $\Q(\sqrt{2}, \sqrt{5})$, $\Q(\sqrt{-2}, \sqrt{5})$, $\Q(\sqrt{3}, \sqrt{5})$, $\Q(\sqrt{-3}, \sqrt{5})$, $\Q(\sqrt{2}, \sqrt{-5})$, $\Q(\sqrt{-2}, \sqrt{-5})$, $\Q(\sqrt{-3}, \sqrt{-5})$, $\Q(\sqrt{3}, \sqrt{-5})$, $\Q(\sqrt{2}, \sqrt{15})$, $\Q(\sqrt{3}, \sqrt{10})$, $\Q(\sqrt{-3}, \sqrt{-10})$, $\Q(\sqrt{-2}, \sqrt{-15})$, $\Q(\sqrt{2}, \sqrt{-15})$, $\Q(\sqrt{-3}, \sqrt{10})$, $\Q(\sqrt{3}, \sqrt{-10})$, $\Q(\sqrt{-2}, \sqrt{15})$, $\Q(\sqrt{2}, \sqrt{-3})$, $\Q(\sqrt{-6}, \sqrt{10})$, $\Q(\sqrt{-6}, \sqrt{-10})$, $\Q(\sqrt{-2}, \sqrt{3})$, $\Q(\sqrt{2}, \sqrt{3})$, $\Q(\sqrt{6}, \sqrt{10})$, $\Q(\sqrt{6}, \sqrt{-10})$, $\Q(\sqrt{-2}, \sqrt{-3})$, $\Q(\zeta_{8})$, $\Q(i, \sqrt{10})$, $\Q(i, \sqrt{15})$, $\Q(\zeta_{12})$, 8.0.3317760000.9, 8.0.40960000.1, 8.0.12960000.1, 8.8.3317760000.1, 8.0.207360000.2, 8.0.207360000.1, 8.0.3317760000.7, 8.0.3317760000.3, 8.0.3317760000.1, 8.0.3317760000.6, 8.0.3317760000.8, 8.0.3317760000.4, 8.0.3317760000.2, $\Q(\zeta_{24})$, 8.0.3317760000.5, 8.0.958832640000.35, 8.8.59927040000.2, 8.0.739840000.6, 8.8.11837440000.1, 8.0.958832640000.21, 8.0.958832640000.65, 8.0.11837440000.5, 8.0.11837440000.9, 8.0.958832640000.41, 8.0.958832640000.20, 8.8.958832640000.3, 8.8.958832640000.5, 8.0.59927040000.32, 8.0.59927040000.35, 8.0.958832640000.11, 8.0.958832640000.26, 8.0.95883264.1, 8.0.59927040000.12, 8.0.958832640000.42, 8.0.1534132224.10, 8.0.1534132224.4, 8.0.958832640000.29, 8.8.958832640000.4, 8.8.1534132224.1, 8.0.1534132224.8, 8.0.958832640000.82, 8.0.11837440000.20, 8.0.18939904.2, 16.0.11007531417600000000.1, 16.0.919360031529369600000000.5, 16.0.140124985753600000000.3, 16.0.919360031529369600000000.7, 16.16.919360031529369600000000.1, 16.0.3591250123161600000000.1, 16.0.919360031529369600000000.2, 16.0.919360031529369600000000.3, 16.0.919360031529369600000000.6, 16.0.919360031529369600000000.1, 16.0.919360031529369600000000.9, 16.0.919360031529369600000000.8, 16.0.919360031529369600000000.4, 16.0.2353561680715186176.2, 16.0.919360031529369600000000.10

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

sage: K.subfields()[1:-1]

gp: L = nfsubfields(K); L[2..length(b)]

magma: L := Subfields(K); L[2..#L];

oscar: subfields(K)[2:end-1]

Sibling fields

Degree 32 siblings:	data not computed
Minimal sibling:	This field is its own minimal sibling

Frobenius cycle types

$p$	$2$	$3$	$5$	$7$	$11$	$13$	$17$	$19$	$23$	$29$	$31$	$37$	$41$	$43$	$47$	$53$	$59$
Cycle type	R	R	R	${\href{/padicField/7.2.0.1}{2} }^{16}$	${\href{/padicField/11.4.0.1}{4} }^{8}$	${\href{/padicField/13.2.0.1}{2} }^{16}$	R	${\href{/padicField/19.2.0.1}{2} }^{16}$	${\href{/padicField/23.2.0.1}{2} }^{16}$	${\href{/padicField/29.4.0.1}{4} }^{8}$	${\href{/padicField/31.2.0.1}{2} }^{16}$	${\href{/padicField/37.4.0.1}{4} }^{8}$	${\href{/padicField/41.2.0.1}{2} }^{16}$	${\href{/padicField/43.2.0.1}{2} }^{16}$	${\href{/padicField/47.2.0.1}{2} }^{16}$	${\href{/padicField/53.2.0.1}{2} }^{16}$	${\href{/padicField/59.2.0.1}{2} }^{16}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Sage:

p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]

\\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Pari:

p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])

// to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7 in Magma:

p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];

# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Oscar:

p = 7; pfac = factor(ideal(ring_of_integers(K), p)); [(e, valuation(norm(pr),p)) for (pr,e) in pfac]

Local algebras for ramified primes

$p$	Label	Polynomial	$e$	$f$	$c$	Galois group	Slope content
$2$ 2	2.8.16.6	$x^{8} + 4 x^{7} + 24 x^{6} + 48 x^{5} + 56 x^{4} + 56 x^{3} + 64 x^{2} + 48 x + 36$	$4$	$2$	$16$	$C_2^3$	$[2, 3]^{2}$
	2.8.16.6	$x^{8} + 4 x^{7} + 24 x^{6} + 48 x^{5} + 56 x^{4} + 56 x^{3} + 64 x^{2} + 48 x + 36$	$4$	$2$	$16$	$C_2^3$	$[2, 3]^{2}$
	2.8.16.6	$x^{8} + 4 x^{7} + 24 x^{6} + 48 x^{5} + 56 x^{4} + 56 x^{3} + 64 x^{2} + 48 x + 36$	$4$	$2$	$16$	$C_2^3$	$[2, 3]^{2}$
	2.8.16.6	$x^{8} + 4 x^{7} + 24 x^{6} + 48 x^{5} + 56 x^{4} + 56 x^{3} + 64 x^{2} + 48 x + 36$	$4$	$2$	$16$	$C_2^3$	$[2, 3]^{2}$
$3$ 3	3.8.4.1	$x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$	$2$	$4$	$4$	$C_4\times C_2$	$[\ ]_{2}^{4}$
	3.8.4.1	$x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$	$2$	$4$	$4$	$C_4\times C_2$	$[\ ]_{2}^{4}$
	3.8.4.1	$x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$	$2$	$4$	$4$	$C_4\times C_2$	$[\ ]_{2}^{4}$
	3.8.4.1	$x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$	$2$	$4$	$4$	$C_4\times C_2$	$[\ ]_{2}^{4}$
$5$ 5	5.8.4.1	$x^{8} + 80 x^{7} + 2428 x^{6} + 33688 x^{5} + 195810 x^{4} + 305952 x^{3} + 870132 x^{2} + 1037416 x + 503089$	$2$	$4$	$4$	$C_4\times C_2$	$[\ ]_{2}^{4}$
	5.8.4.1	$x^{8} + 80 x^{7} + 2428 x^{6} + 33688 x^{5} + 195810 x^{4} + 305952 x^{3} + 870132 x^{2} + 1037416 x + 503089$	$2$	$4$	$4$	$C_4\times C_2$	$[\ ]_{2}^{4}$
	5.8.4.1	$x^{8} + 80 x^{7} + 2428 x^{6} + 33688 x^{5} + 195810 x^{4} + 305952 x^{3} + 870132 x^{2} + 1037416 x + 503089$	$2$	$4$	$4$	$C_4\times C_2$	$[\ ]_{2}^{4}$
	5.8.4.1	$x^{8} + 80 x^{7} + 2428 x^{6} + 33688 x^{5} + 195810 x^{4} + 305952 x^{3} + 870132 x^{2} + 1037416 x + 503089$	$2$	$4$	$4$	$C_4\times C_2$	$[\ ]_{2}^{4}$
$17$ 17	17.2.0.1	$x^{2} + 16 x + 3$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
	17.2.0.1	$x^{2} + 16 x + 3$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
	17.2.0.1	$x^{2} + 16 x + 3$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
	17.2.0.1	$x^{2} + 16 x + 3$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
	17.2.0.1	$x^{2} + 16 x + 3$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
	17.2.0.1	$x^{2} + 16 x + 3$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
	17.2.0.1	$x^{2} + 16 x + 3$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
	17.2.0.1	$x^{2} + 16 x + 3$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
	17.4.2.1	$x^{4} + 338 x^{3} + 31049 x^{2} + 420472 x + 123735$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$
	17.4.2.1	$x^{4} + 338 x^{3} + 31049 x^{2} + 420472 x + 123735$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$
	17.4.2.1	$x^{4} + 338 x^{3} + 31049 x^{2} + 420472 x + 123735$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$
	17.4.2.1	$x^{4} + 338 x^{3} + 31049 x^{2} + 420472 x + 123735$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$

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