LMFDB - Number field 32.0.126945864674316252168838917102904812361796913463296.1

Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^32 + 2*x^30 + x^28 - 6*x^26 - 28*x^24 - 58*x^22 + 9*x^20 + 230*x^18 + 593*x^16 + 920*x^14 + 144*x^12 - 3712*x^10 - 7168*x^8 - 6144*x^6 + 4096*x^4 + 32768*x^2 + 65536)

gp: K = bnfinit(y^32 + 2*y^30 + y^28 - 6*y^26 - 28*y^24 - 58*y^22 + 9*y^20 + 230*y^18 + 593*y^16 + 920*y^14 + 144*y^12 - 3712*y^10 - 7168*y^8 - 6144*y^6 + 4096*y^4 + 32768*y^2 + 65536, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^32 + 2*x^30 + x^28 - 6*x^26 - 28*x^24 - 58*x^22 + 9*x^20 + 230*x^18 + 593*x^16 + 920*x^14 + 144*x^12 - 3712*x^10 - 7168*x^8 - 6144*x^6 + 4096*x^4 + 32768*x^2 + 65536);

oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^32 + 2*x^30 + x^28 - 6*x^26 - 28*x^24 - 58*x^22 + 9*x^20 + 230*x^18 + 593*x^16 + 920*x^14 + 144*x^12 - 3712*x^10 - 7168*x^8 - 6144*x^6 + 4096*x^4 + 32768*x^2 + 65536)

$ x^{32} + 2 x^{30} + x^{28} - 6 x^{26} - 28 x^{24} - 58 x^{22} + 9 x^{20} + 230 x^{18} + 593 x^{16} + \cdots + 65536 $ x^32 + 2*x^30 + x^28 - 6*x^26 - 28*x^24 - 58*x^22 + 9*x^20 + 230*x^18 + 593*x^16 + 920*x^14 + 144*x^12 - 3712*x^10 - 7168*x^8 - 6144*x^6 + 4096*x^4 + 32768*x^2 + 65536

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:	$126945864674316252168838917102904812361796913463296$ 126945864674316252168838917102904812361796913463296 $\medspace = 2^{32}\cdot 3^{16}\cdot 137^{4}\cdot 1087^{8}$	sage: K.disc() gp: K.disc magma: OK := Integers(K); Discriminant(OK); oscar: OK = ring_of_integers(K); discriminant(OK)
Root discriminant:	$36.79$	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:	$2\cdot 3^{1/2}137^{1/2}1087^{1/2}\approx 1336.7976660661852$
Ramified primes:	$2$, $3$, $137$, $1087$ 2, 3, 137, 1087	sage: K.disc().support() gp: factor(abs(K.disc))[,1]~ magma: PrimeDivisors(Discriminant(OK)); oscar: prime_divisors(discriminant((OK)))
Discriminant root field:	$\Q$
$\card{ \Aut(K/\Q) }$:	$8$	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 $a$)

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2}a^{7}-\frac{1}{2}a$, $\frac{1}{2}a^{8}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{3}$, $\frac{1}{4}a^{10}-\frac{1}{4}a^{8}-\frac{1}{4}a^{4}-\frac{1}{2}a^{3}+\frac{1}{4}a^{2}-\frac{1}{2}a$, $\frac{1}{4}a^{11}-\frac{1}{4}a^{9}-\frac{1}{4}a^{5}-\frac{1}{2}a^{4}+\frac{1}{4}a^{3}-\frac{1}{2}a^{2}$, $\frac{1}{4}a^{12}-\frac{1}{4}a^{8}-\frac{1}{4}a^{6}-\frac{1}{2}a^{5}+\frac{1}{4}a^{2}-\frac{1}{2}a$, $\frac{1}{4}a^{13}-\frac{1}{4}a^{9}-\frac{1}{4}a^{7}-\frac{1}{2}a^{6}+\frac{1}{4}a^{3}-\frac{1}{2}a^{2}$, $\frac{1}{4}a^{14}-\frac{1}{4}a^{2}$, $\frac{1}{8}a^{15}-\frac{1}{4}a^{9}-\frac{1}{4}a^{8}-\frac{1}{2}a^{5}-\frac{3}{8}a^{3}+\frac{1}{4}a^{2}-\frac{1}{2}a$, $\frac{1}{8}a^{16}-\frac{1}{4}a^{9}-\frac{1}{4}a^{8}-\frac{1}{2}a^{6}+\frac{3}{8}a^{4}-\frac{1}{4}a^{3}-\frac{1}{4}a^{2}-\frac{1}{2}a$, $\frac{1}{8}a^{17}-\frac{1}{4}a^{9}-\frac{1}{4}a^{8}+\frac{3}{8}a^{5}-\frac{1}{2}a^{4}+\frac{1}{4}a^{3}-\frac{1}{4}a^{2}$, $\frac{1}{16}a^{18}-\frac{1}{16}a^{16}-\frac{1}{8}a^{12}-\frac{1}{8}a^{10}+\frac{1}{16}a^{6}-\frac{1}{16}a^{4}-\frac{1}{2}a^{3}+\frac{1}{4}a^{2}$, $\frac{1}{16}a^{19}-\frac{1}{16}a^{17}-\frac{1}{8}a^{13}-\frac{1}{8}a^{11}+\frac{1}{16}a^{7}-\frac{1}{16}a^{5}-\frac{1}{2}a^{4}+\frac{1}{4}a^{3}$, $\frac{1}{16}a^{20}-\frac{1}{16}a^{16}-\frac{1}{8}a^{14}-\frac{1}{8}a^{10}-\frac{3}{16}a^{8}-\frac{1}{4}a^{6}+\frac{3}{16}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{32}a^{21}-\frac{1}{32}a^{17}-\frac{1}{16}a^{15}+\frac{1}{16}a^{11}-\frac{7}{32}a^{9}-\frac{1}{8}a^{7}-\frac{1}{2}a^{6}-\frac{1}{32}a^{5}-\frac{1}{2}a^{4}+\frac{3}{8}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{128}a^{22}+\frac{1}{64}a^{20}+\frac{1}{128}a^{18}-\frac{3}{64}a^{16}-\frac{3}{32}a^{14}+\frac{3}{64}a^{12}+\frac{9}{128}a^{10}-\frac{1}{4}a^{9}+\frac{3}{64}a^{8}-\frac{47}{128}a^{6}-\frac{5}{16}a^{4}+\frac{1}{4}a^{3}-\frac{1}{4}a^{2}-\frac{1}{2}$, $\frac{1}{256}a^{23}+\frac{1}{128}a^{21}+\frac{1}{256}a^{19}+\frac{5}{128}a^{17}-\frac{3}{64}a^{15}+\frac{3}{128}a^{13}-\frac{23}{256}a^{11}-\frac{29}{128}a^{9}-\frac{1}{4}a^{8}+\frac{17}{256}a^{7}-\frac{1}{2}a^{6}+\frac{5}{32}a^{5}-\frac{1}{8}a^{3}+\frac{1}{4}a^{2}+\frac{1}{4}a$, $\frac{1}{3072}a^{24}-\frac{5}{1536}a^{22}-\frac{7}{3072}a^{20}+\frac{7}{1536}a^{18}-\frac{1}{16}a^{17}+\frac{5}{256}a^{16}-\frac{1}{16}a^{15}-\frac{101}{1536}a^{14}+\frac{1}{3072}a^{12}-\frac{1}{8}a^{11}+\frac{109}{1536}a^{10}-\frac{1}{8}a^{9}-\frac{77}{1024}a^{8}-\frac{181}{768}a^{6}+\frac{3}{16}a^{5}-\frac{11}{24}a^{4}-\frac{1}{16}a^{3}+\frac{11}{48}a^{2}-\frac{1}{4}a+\frac{1}{12}$, $\frac{1}{6144}a^{25}-\frac{5}{3072}a^{23}-\frac{7}{6144}a^{21}-\frac{1}{32}a^{20}+\frac{7}{3072}a^{19}-\frac{27}{512}a^{17}+\frac{1}{32}a^{16}-\frac{101}{3072}a^{15}+\frac{1}{16}a^{14}-\frac{767}{6144}a^{13}-\frac{275}{3072}a^{11}-\frac{1}{16}a^{10}-\frac{77}{2048}a^{9}-\frac{1}{32}a^{8}-\frac{373}{1536}a^{7}-\frac{3}{8}a^{6}+\frac{11}{24}a^{5}-\frac{15}{32}a^{4}-\frac{13}{96}a^{3}-\frac{1}{8}a^{2}-\frac{11}{24}a$, $\frac{1}{12288}a^{26}-\frac{1}{6144}a^{24}+\frac{3}{4096}a^{22}-\frac{1}{64}a^{21}-\frac{39}{2048}a^{20}-\frac{1}{32}a^{19}+\frac{67}{3072}a^{18}+\frac{3}{64}a^{17}+\frac{235}{6144}a^{16}+\frac{1}{32}a^{15}+\frac{305}{12288}a^{14}-\frac{1}{16}a^{13}+\frac{401}{6144}a^{12}+\frac{1}{32}a^{11}+\frac{841}{12288}a^{10}+\frac{15}{64}a^{9}+\frac{749}{3072}a^{8}-\frac{3}{32}a^{7}-\frac{5}{64}a^{6}-\frac{29}{64}a^{5}-\frac{23}{64}a^{4}+\frac{1}{16}a^{3}+\frac{11}{48}a^{2}-\frac{1}{2}a-\frac{1}{3}$, $\frac{1}{24576}a^{27}-\frac{1}{12288}a^{25}+\frac{3}{8192}a^{23}-\frac{39}{4096}a^{21}-\frac{1}{32}a^{20}-\frac{125}{6144}a^{19}-\frac{149}{12288}a^{17}+\frac{1}{32}a^{16}+\frac{305}{24576}a^{15}-\frac{1}{16}a^{14}+\frac{1169}{12288}a^{13}+\frac{2377}{24576}a^{11}-\frac{1}{16}a^{10}-\frac{19}{6144}a^{9}+\frac{7}{32}a^{8}+\frac{23}{128}a^{7}+\frac{1}{8}a^{6}-\frac{11}{128}a^{5}-\frac{15}{32}a^{4}-\frac{13}{96}a^{3}-\frac{1}{4}a^{2}-\frac{1}{6}a-\frac{1}{2}$, $\frac{1}{28999680}a^{28}+\frac{35}{966656}a^{26}-\frac{181}{9666560}a^{24}-\frac{1}{512}a^{23}-\frac{4699}{2899968}a^{22}+\frac{3}{256}a^{21}+\frac{74667}{2416640}a^{20}-\frac{1}{512}a^{19}-\frac{93951}{4833280}a^{18}+\frac{7}{256}a^{17}+\frac{18199}{1933312}a^{16}-\frac{1}{128}a^{15}-\frac{1108753}{14499840}a^{14}-\frac{3}{256}a^{13}-\frac{249187}{5799936}a^{12}+\frac{39}{512}a^{11}+\frac{58423}{604160}a^{10}-\frac{31}{256}a^{9}-\frac{39043}{226560}a^{8}-\frac{49}{512}a^{7}+\frac{13241}{45312}a^{6}+\frac{3}{32}a^{5}-\frac{2717}{7080}a^{4}+\frac{3}{8}a^{3}+\frac{505}{1416}a^{2}+\frac{1}{8}a-\frac{1471}{7080}$, $\frac{1}{57999360}a^{29}+\frac{35}{1933312}a^{27}-\frac{181}{19333120}a^{25}-\frac{4699}{5799936}a^{23}-\frac{1}{256}a^{22}+\frac{74667}{4833280}a^{21}+\frac{3}{128}a^{20}-\frac{93951}{9666560}a^{19}-\frac{1}{256}a^{18}-\frac{223465}{3866624}a^{17}+\frac{7}{128}a^{16}+\frac{703727}{28999680}a^{15}-\frac{1}{64}a^{14}-\frac{249187}{11599872}a^{13}-\frac{3}{128}a^{12}+\frac{58423}{1208320}a^{11}-\frac{25}{256}a^{10}+\frac{74237}{453120}a^{9}-\frac{31}{128}a^{8}+\frac{13241}{90624}a^{7}-\frac{49}{256}a^{6}+\frac{427}{3540}a^{5}-\frac{1}{16}a^{4}-\frac{68}{177}a^{3}+\frac{1}{4}a^{2}+\frac{2069}{14160}a+\frac{1}{4}$, $\frac{1}{90710999040}a^{30}+\frac{1}{177864704}a^{28}+\frac{2736457}{90710999040}a^{26}+\frac{504175}{9071099904}a^{24}-\frac{1}{512}a^{23}+\frac{53007731}{22677749760}a^{22}+\frac{3}{256}a^{21}+\frac{78476891}{2667970560}a^{20}-\frac{1}{512}a^{19}+\frac{84065599}{6047399936}a^{18}-\frac{9}{256}a^{17}-\frac{414539621}{15118499840}a^{16}+\frac{7}{128}a^{15}+\frac{2214508597}{18142199808}a^{14}-\frac{3}{256}a^{13}+\frac{753549821}{22677749760}a^{12}-\frac{25}{512}a^{11}+\frac{376427}{5210880}a^{10}-\frac{63}{256}a^{9}-\frac{30484535}{141735936}a^{8}+\frac{79}{512}a^{7}-\frac{17503027}{59056640}a^{6}+\frac{9}{32}a^{5}-\frac{1419301}{4429248}a^{4}-\frac{5}{16}a^{3}-\frac{194301}{434240}a^{2}+\frac{1}{8}a+\frac{87107}{369104}$, $\frac{1}{181421998080}a^{31}+\frac{1}{355729408}a^{29}+\frac{2736457}{181421998080}a^{27}+\frac{504175}{18142199808}a^{25}+\frac{53007731}{45355499520}a^{23}-\frac{1}{256}a^{22}+\frac{78476891}{5335941120}a^{21}-\frac{1}{128}a^{20}-\frac{293896897}{12094799872}a^{19}-\frac{1}{256}a^{18}+\frac{530366619}{30236999680}a^{17}-\frac{5}{128}a^{16}-\frac{53266379}{36284399616}a^{15}-\frac{5}{64}a^{14}+\frac{3588268541}{45355499520}a^{13}-\frac{3}{128}a^{12}+\frac{1027787}{10421760}a^{11}+\frac{23}{256}a^{10}+\frac{4949449}{283471872}a^{9}+\frac{29}{128}a^{8}-\frac{21194067}{118113280}a^{7}+\frac{111}{256}a^{6}-\frac{3357097}{8858496}a^{5}-\frac{5}{32}a^{4}+\frac{294219}{868480}a^{3}-\frac{1}{4}a^{2}-\frac{97445}{738208}a+\frac{1}{4}$ 1, a, a^2, a^3, a^4, a^5, a^6, 1/2*a^7 - 1/2*a, 1/2*a^8 - 1/2*a^2, 1/2*a^9 - 1/2*a^3, 1/4*a^10 - 1/4*a^8 - 1/4*a^4 - 1/2*a^3 + 1/4*a^2 - 1/2*a, 1/4*a^11 - 1/4*a^9 - 1/4*a^5 - 1/2*a^4 + 1/4*a^3 - 1/2*a^2, 1/4*a^12 - 1/4*a^8 - 1/4*a^6 - 1/2*a^5 + 1/4*a^2 - 1/2*a, 1/4*a^13 - 1/4*a^9 - 1/4*a^7 - 1/2*a^6 + 1/4*a^3 - 1/2*a^2, 1/4*a^14 - 1/4*a^2, 1/8*a^15 - 1/4*a^9 - 1/4*a^8 - 1/2*a^5 - 3/8*a^3 + 1/4*a^2 - 1/2*a, 1/8*a^16 - 1/4*a^9 - 1/4*a^8 - 1/2*a^6 + 3/8*a^4 - 1/4*a^3 - 1/4*a^2 - 1/2*a, 1/8*a^17 - 1/4*a^9 - 1/4*a^8 + 3/8*a^5 - 1/2*a^4 + 1/4*a^3 - 1/4*a^2, 1/16*a^18 - 1/16*a^16 - 1/8*a^12 - 1/8*a^10 + 1/16*a^6 - 1/16*a^4 - 1/2*a^3 + 1/4*a^2, 1/16*a^19 - 1/16*a^17 - 1/8*a^13 - 1/8*a^11 + 1/16*a^7 - 1/16*a^5 - 1/2*a^4 + 1/4*a^3, 1/16*a^20 - 1/16*a^16 - 1/8*a^14 - 1/8*a^10 - 3/16*a^8 - 1/4*a^6 + 3/16*a^4 - 1/2*a^3 - 1/2*a^2 - 1/2*a, 1/32*a^21 - 1/32*a^17 - 1/16*a^15 + 1/16*a^11 - 7/32*a^9 - 1/8*a^7 - 1/2*a^6 - 1/32*a^5 - 1/2*a^4 + 3/8*a^3 - 1/2*a^2 - 1/2*a, 1/128*a^22 + 1/64*a^20 + 1/128*a^18 - 3/64*a^16 - 3/32*a^14 + 3/64*a^12 + 9/128*a^10 - 1/4*a^9 + 3/64*a^8 - 47/128*a^6 - 5/16*a^4 + 1/4*a^3 - 1/4*a^2 - 1/2, 1/256*a^23 + 1/128*a^21 + 1/256*a^19 + 5/128*a^17 - 3/64*a^15 + 3/128*a^13 - 23/256*a^11 - 29/128*a^9 - 1/4*a^8 + 17/256*a^7 - 1/2*a^6 + 5/32*a^5 - 1/8*a^3 + 1/4*a^2 + 1/4*a, 1/3072*a^24 - 5/1536*a^22 - 7/3072*a^20 + 7/1536*a^18 - 1/16*a^17 + 5/256*a^16 - 1/16*a^15 - 101/1536*a^14 + 1/3072*a^12 - 1/8*a^11 + 109/1536*a^10 - 1/8*a^9 - 77/1024*a^8 - 181/768*a^6 + 3/16*a^5 - 11/24*a^4 - 1/16*a^3 + 11/48*a^2 - 1/4*a + 1/12, 1/6144*a^25 - 5/3072*a^23 - 7/6144*a^21 - 1/32*a^20 + 7/3072*a^19 - 27/512*a^17 + 1/32*a^16 - 101/3072*a^15 + 1/16*a^14 - 767/6144*a^13 - 275/3072*a^11 - 1/16*a^10 - 77/2048*a^9 - 1/32*a^8 - 373/1536*a^7 - 3/8*a^6 + 11/24*a^5 - 15/32*a^4 - 13/96*a^3 - 1/8*a^2 - 11/24*a, 1/12288*a^26 - 1/6144*a^24 + 3/4096*a^22 - 1/64*a^21 - 39/2048*a^20 - 1/32*a^19 + 67/3072*a^18 + 3/64*a^17 + 235/6144*a^16 + 1/32*a^15 + 305/12288*a^14 - 1/16*a^13 + 401/6144*a^12 + 1/32*a^11 + 841/12288*a^10 + 15/64*a^9 + 749/3072*a^8 - 3/32*a^7 - 5/64*a^6 - 29/64*a^5 - 23/64*a^4 + 1/16*a^3 + 11/48*a^2 - 1/2*a - 1/3, 1/24576*a^27 - 1/12288*a^25 + 3/8192*a^23 - 39/4096*a^21 - 1/32*a^20 - 125/6144*a^19 - 149/12288*a^17 + 1/32*a^16 + 305/24576*a^15 - 1/16*a^14 + 1169/12288*a^13 + 2377/24576*a^11 - 1/16*a^10 - 19/6144*a^9 + 7/32*a^8 + 23/128*a^7 + 1/8*a^6 - 11/128*a^5 - 15/32*a^4 - 13/96*a^3 - 1/4*a^2 - 1/6*a - 1/2, 1/28999680*a^28 + 35/966656*a^26 - 181/9666560*a^24 - 1/512*a^23 - 4699/2899968*a^22 + 3/256*a^21 + 74667/2416640*a^20 - 1/512*a^19 - 93951/4833280*a^18 + 7/256*a^17 + 18199/1933312*a^16 - 1/128*a^15 - 1108753/14499840*a^14 - 3/256*a^13 - 249187/5799936*a^12 + 39/512*a^11 + 58423/604160*a^10 - 31/256*a^9 - 39043/226560*a^8 - 49/512*a^7 + 13241/45312*a^6 + 3/32*a^5 - 2717/7080*a^4 + 3/8*a^3 + 505/1416*a^2 + 1/8*a - 1471/7080, 1/57999360*a^29 + 35/1933312*a^27 - 181/19333120*a^25 - 4699/5799936*a^23 - 1/256*a^22 + 74667/4833280*a^21 + 3/128*a^20 - 93951/9666560*a^19 - 1/256*a^18 - 223465/3866624*a^17 + 7/128*a^16 + 703727/28999680*a^15 - 1/64*a^14 - 249187/11599872*a^13 - 3/128*a^12 + 58423/1208320*a^11 - 25/256*a^10 + 74237/453120*a^9 - 31/128*a^8 + 13241/90624*a^7 - 49/256*a^6 + 427/3540*a^5 - 1/16*a^4 - 68/177*a^3 + 1/4*a^2 + 2069/14160*a + 1/4, 1/90710999040*a^30 + 1/177864704*a^28 + 2736457/90710999040*a^26 + 504175/9071099904*a^24 - 1/512*a^23 + 53007731/22677749760*a^22 + 3/256*a^21 + 78476891/2667970560*a^20 - 1/512*a^19 + 84065599/6047399936*a^18 - 9/256*a^17 - 414539621/15118499840*a^16 + 7/128*a^15 + 2214508597/18142199808*a^14 - 3/256*a^13 + 753549821/22677749760*a^12 - 25/512*a^11 + 376427/5210880*a^10 - 63/256*a^9 - 30484535/141735936*a^8 + 79/512*a^7 - 17503027/59056640*a^6 + 9/32*a^5 - 1419301/4429248*a^4 - 5/16*a^3 - 194301/434240*a^2 + 1/8*a + 87107/369104, 1/181421998080*a^31 + 1/355729408*a^29 + 2736457/181421998080*a^27 + 504175/18142199808*a^25 + 53007731/45355499520*a^23 - 1/256*a^22 + 78476891/5335941120*a^21 - 1/128*a^20 - 293896897/12094799872*a^19 - 1/256*a^18 + 530366619/30236999680*a^17 - 5/128*a^16 - 53266379/36284399616*a^15 - 5/64*a^14 + 3588268541/45355499520*a^13 - 3/128*a^12 + 1027787/10421760*a^11 + 23/256*a^10 + 4949449/283471872*a^9 + 29/128*a^8 - 21194067/118113280*a^7 + 111/256*a^6 - 3357097/8858496*a^5 - 5/32*a^4 + 294219/868480*a^3 - 1/4*a^2 - 97445/738208*a + 1/4

sage: K.integral_basis()

gp: K.zk

magma: IntegralBasis(K);

oscar: basis(OK)

Monogenic:		No
Index:		Not computed
Inessential primes:		$2$, $3$

Class group and class number

$C_{12}$, which has order $12$ (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{1269343}{90710999040} a^{31} + \frac{85201}{2667970560} a^{29} + \frac{6679831}{90710999040} a^{27} + \frac{6902069}{45355499520} a^{25} - \frac{4480787}{22677749760} a^{23} - \frac{1808443}{2667970560} a^{21} - \frac{60743963}{30236999680} a^{19} - \frac{22762683}{15118499840} a^{17} + \frac{125489437}{30236999680} a^{15} + \frac{137011201}{7559249920} a^{13} + \frac{101003}{5210880} a^{11} + \frac{18115543}{708679680} a^{9} + \frac{1479937}{177169920} a^{7} - \frac{2666533}{22146240} a^{5} - \frac{148409}{1302720} a^{3} + \frac{383579}{5536560} a $ (1269343)/(90710999040)a^(31) + (85201)/(2667970560)a^(29) + (6679831)/(90710999040)a^(27) + (6902069)/(45355499520)a^(25) - (4480787)/(22677749760)a^(23) - (1808443)/(2667970560)a^(21) - (60743963)/(30236999680)a^(19) - (22762683)/(15118499840)a^(17) + (125489437)/(30236999680)a^(15) + (137011201)/(7559249920)a^(13) + (101003)/(5210880)a^(11) + (18115543)/(708679680)a^(9) + (1479937)/(177169920)a^(7) - (2666533)/(22146240)a^(5) - (148409)/(1302720)a^(3) + (383579)/(5536560)a (order $12$)	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:	$\frac{1269343}{90710999040}a^{31}+\frac{85201}{2667970560}a^{29}+\frac{6679831}{90710999040}a^{27}+\frac{6902069}{45355499520}a^{25}-\frac{4480787}{22677749760}a^{23}-\frac{1808443}{2667970560}a^{21}-\frac{60743963}{30236999680}a^{19}-\frac{22762683}{15118499840}a^{17}+\frac{125489437}{30236999680}a^{15}+\frac{137011201}{7559249920}a^{13}+\frac{101003}{5210880}a^{11}+\frac{18115543}{708679680}a^{9}+\frac{1479937}{177169920}a^{7}-\frac{2666533}{22146240}a^{5}-\frac{148409}{1302720}a^{3}+\frac{383579}{5536560}a-1$, $\frac{2003171}{181421998080}a^{31}+\frac{102503}{4535549952}a^{30}-\frac{50143}{5335941120}a^{29}+\frac{1235}{33349632}a^{28}+\frac{2917209}{60473999360}a^{27}+\frac{505675}{4535549952}a^{26}+\frac{9704053}{90710999040}a^{25}+\frac{3725}{47245312}a^{24}+\frac{970367}{15118499840}a^{23}-\frac{52677}{94490624}a^{22}-\frac{1247551}{5335941120}a^{21}-\frac{96391}{133398528}a^{20}-\frac{17133853}{181421998080}a^{19}-\frac{1938871}{1511849984}a^{18}-\frac{154874873}{90710999040}a^{17}+\frac{1786465}{1133887488}a^{16}+\frac{234489}{60473999360}a^{15}+\frac{37028387}{4535549952}a^{14}+\frac{81595147}{15118499840}a^{13}+\frac{37462121}{2267774976}a^{12}+\frac{2984051}{166748160}a^{11}+\frac{579799}{16674816}a^{10}+\frac{19895587}{472453120}a^{9}+\frac{439169}{17716992}a^{8}+\frac{16633009}{354339840}a^{7}-\frac{162149}{2214624}a^{6}-\frac{3199763}{22146240}a^{5}-\frac{560113}{4429248}a^{4}-\frac{185713}{2605440}a^{3}-\frac{2531}{21712}a^{2}-\frac{4032907}{11073120}a+\frac{32255}{553656}$, $\frac{2294729}{90710999040}a^{31}-\frac{351443}{18142199808}a^{30}+\frac{63851}{889323520}a^{29}-\frac{38317}{533594112}a^{28}+\frac{2836371}{30236999680}a^{27}+\frac{372757}{18142199808}a^{26}-\frac{5851523}{45355499520}a^{25}+\frac{883861}{3023699968}a^{24}-\frac{16176931}{22677749760}a^{23}+\frac{3517255}{4535549952}a^{22}-\frac{3831109}{2667970560}a^{21}+\frac{122389}{177864704}a^{20}-\frac{10000207}{90710999040}a^{19}-\frac{2461091}{18142199808}a^{18}+\frac{163089283}{45355499520}a^{17}-\frac{22434353}{3023699968}a^{16}+\frac{1120934473}{90710999040}a^{15}-\frac{106884523}{18142199808}a^{14}+\frac{214184777}{11338874880}a^{13}-\frac{9589797}{1511849984}a^{12}-\frac{1571477}{166748160}a^{11}+\frac{200459}{8337408}a^{10}-\frac{14710463}{354339840}a^{9}+\frac{1715359}{47245312}a^{8}-\frac{16337059}{177169920}a^{7}+\frac{2797825}{35433984}a^{6}-\frac{238521}{14764160}a^{5}-\frac{156605}{4429248}a^{4}+\frac{15881}{434240}a^{3}-\frac{63875}{260544}a^{2}+\frac{883363}{1384140}a-\frac{104625}{369104}$, $\frac{29537}{18142199808}a^{31}+\frac{181449}{6047399936}a^{30}-\frac{29825}{533594112}a^{29}+\frac{12151}{177864704}a^{28}-\frac{538461}{6047399936}a^{27}+\frac{1457587}{18142199808}a^{26}+\frac{1342747}{9071099904}a^{25}-\frac{4294823}{9071099904}a^{24}+\frac{799843}{4535549952}a^{23}-\frac{2848303}{4535549952}a^{22}+\frac{385979}{533594112}a^{21}-\frac{345703}{533594112}a^{20}+\frac{5831633}{18142199808}a^{19}+\frac{1422649}{6047399936}a^{18}+\frac{4705921}{9071099904}a^{17}+\frac{61086043}{9071099904}a^{16}-\frac{99216343}{18142199808}a^{15}+\frac{241065331}{18142199808}a^{14}-\frac{14971195}{4535549952}a^{13}+\frac{48470135}{4535549952}a^{12}-\frac{14767}{1042176}a^{11}-\frac{44867}{2779136}a^{10}-\frac{3349079}{141735936}a^{9}-\frac{11462555}{141735936}a^{8}+\frac{363183}{5905664}a^{7}-\frac{4216685}{35433984}a^{6}+\frac{283717}{4429248}a^{5}-\frac{51685}{4429248}a^{4}-\frac{30407}{260544}a^{3}+\frac{222119}{260544}a^{2}-\frac{7095}{369104}a-\frac{99111}{369104}$, $\frac{23953}{10671882240}a^{31}+\frac{14385}{3023699968}a^{30}+\frac{10531}{5335941120}a^{29}+\frac{2635}{266797056}a^{28}+\frac{50147}{3557294080}a^{27}-\frac{392317}{9071099904}a^{26}-\frac{436273}{5335941120}a^{25}-\frac{58475}{1511849984}a^{24}-\frac{839257}{2667970560}a^{23}-\frac{258665}{2267774976}a^{22}+\frac{216847}{1067188224}a^{21}-\frac{108487}{266797056}a^{20}+\frac{9389233}{10671882240}a^{19}-\frac{968885}{9071099904}a^{18}+\frac{25982261}{5335941120}a^{17}+\frac{3589025}{4535549952}a^{16}+\frac{62443273}{10671882240}a^{15}-\frac{2330573}{9071099904}a^{14}-\frac{1874777}{2667970560}a^{13}+\frac{3025775}{1133887488}a^{12}-\frac{7345111}{166748160}a^{11}+\frac{7105}{4168704}a^{10}-\frac{4047661}{83374080}a^{9}+\frac{217033}{70867968}a^{8}-\frac{397871}{6947840}a^{7}-\frac{170495}{8858496}a^{6}+\frac{166319}{651360}a^{5}-\frac{5945}{369104}a^{4}+\frac{824677}{2605440}a^{3}-\frac{22109}{130272}a^{2}+\frac{171489}{217120}a-\frac{268693}{276828}$, $\frac{1025357}{90710999040}a^{31}+\frac{35639}{1067188224}a^{30}+\frac{13483}{533594112}a^{29}+\frac{25945}{533594112}a^{28}+\frac{2220389}{90710999040}a^{27}+\frac{39599}{1067188224}a^{26}+\frac{759623}{9071099904}a^{25}-\frac{67571}{533594112}a^{24}-\frac{2816413}{22677749760}a^{23}-\frac{121291}{266797056}a^{22}-\frac{881393}{2667970560}a^{21}-\frac{32803}{533594112}a^{20}-\frac{7917181}{6047399936}a^{19}+\frac{38125}{355729408}a^{18}-\frac{10996997}{15118499840}a^{17}+\frac{660413}{177864704}a^{16}+\frac{21447915}{6047399936}a^{15}+\frac{262949}{355729408}a^{14}+\frac{80343729}{7559249920}a^{13}+\frac{635393}{88932352}a^{12}+\frac{2713583}{166748160}a^{11}+\frac{120479}{16674816}a^{10}+\frac{2627437}{141735936}a^{9}+\frac{415589}{8337408}a^{8}+\frac{425273}{177169920}a^{7}-\frac{77263}{2084352}a^{6}-\frac{467639}{8858496}a^{5}+\frac{10829}{130272}a^{4}-\frac{42811}{1302720}a^{3}-\frac{40829}{260544}a^{2}-\frac{35689}{1107312}a+\frac{5611}{65136}$, $\frac{773309}{36284399616}a^{31}+\frac{571361}{45355499520}a^{30}+\frac{124689}{1778647040}a^{29}+\frac{757}{444661760}a^{28}+\frac{1588085}{36284399616}a^{27}-\frac{7536823}{45355499520}a^{26}-\frac{17504497}{90710999040}a^{25}-\frac{12873061}{22677749760}a^{24}-\frac{2856005}{9071099904}a^{23}-\frac{8472229}{11338874880}a^{22}-\frac{3583237}{5335941120}a^{21}-\frac{120353}{266797056}a^{20}+\frac{55039259}{60473999360}a^{19}+\frac{128869201}{45355499520}a^{18}+\frac{38448379}{6047399936}a^{17}+\frac{99735979}{7559249920}a^{16}+\frac{1799627177}{181421998080}a^{15}+\frac{1046867081}{45355499520}a^{14}-\frac{38749685}{9071099904}a^{13}-\frac{26453193}{3779624960}a^{12}-\frac{75449}{2826240}a^{11}-\frac{1619519}{27791360}a^{10}-\frac{54574639}{1417359360}a^{9}-\frac{32345761}{177169920}a^{8}-\frac{1174967}{23622656}a^{7}-\frac{25824271}{88584960}a^{6}+\frac{3445487}{22146240}a^{5}-\frac{294477}{7382080}a^{4}+\frac{112479}{173696}a^{3}+\frac{603107}{651360}a^{2}+\frac{3139881}{3691040}a+\frac{1969109}{2768280}$, $\frac{10135}{788791296}a^{31}+\frac{5791}{231997440}a^{30}-\frac{4619}{115998720}a^{29}-\frac{1919}{115998720}a^{28}-\frac{102065}{788791296}a^{27}-\frac{13731}{77332480}a^{26}+\frac{48169}{1971978240}a^{25}-\frac{98923}{115998720}a^{24}+\frac{15127}{65732608}a^{23}-\frac{64099}{57999360}a^{22}+\frac{8929}{115998720}a^{21}+\frac{97301}{115998720}a^{20}+\frac{2994017}{1314652160}a^{19}+\frac{1403503}{231997440}a^{18}+\frac{264405}{131465216}a^{17}+\frac{1756847}{115998720}a^{16}-\frac{27530869}{3943956480}a^{15}+\frac{6992983}{231997440}a^{14}-\frac{2823017}{197197824}a^{13}-\frac{385709}{57999360}a^{12}-\frac{51361}{7249920}a^{11}-\frac{449599}{3624960}a^{10}-\frac{153557}{5135360}a^{9}-\frac{360091}{1812480}a^{8}+\frac{9823}{192576}a^{7}-\frac{49497}{151040}a^{6}+\frac{126723}{641920}a^{5}-\frac{2837}{28320}a^{4}+\frac{1529}{11328}a^{3}+\frac{56419}{56640}a^{2}-\frac{107971}{240720}a+\frac{11009}{4720}$, $\frac{969489}{60473999360}a^{31}-\frac{243583}{45355499520}a^{30}+\frac{206501}{5335941120}a^{29}+\frac{171}{88932352}a^{28}-\frac{14155861}{181421998080}a^{27}-\frac{3927551}{45355499520}a^{26}-\frac{5871597}{30236999680}a^{25}+\frac{1194701}{4535549952}a^{24}-\frac{2159141}{15118499840}a^{23}+\frac{3296579}{3779624960}a^{22}-\frac{2813119}{5335941120}a^{21}+\frac{511569}{444661760}a^{20}+\frac{190655011}{181421998080}a^{19}-\frac{1012315}{9071099904}a^{18}+\frac{640679819}{90710999040}a^{17}+\frac{26067959}{22677749760}a^{16}+\frac{1152736811}{181421998080}a^{15}-\frac{213358723}{9071099904}a^{14}-\frac{292752133}{45355499520}a^{13}-\frac{54309121}{1417359360}a^{12}-\frac{139339}{10421760}a^{11}+\frac{999}{6947840}a^{10}-\frac{23890749}{472453120}a^{9}+\frac{1169269}{23622656}a^{8}-\frac{10175369}{118113280}a^{7}+\frac{7374163}{88584960}a^{6}+\frac{2740717}{44292480}a^{5}+\frac{152233}{276828}a^{4}+\frac{1312559}{2605440}a^{3}+\frac{39383}{217120}a^{2}+\frac{892713}{3691040}a-\frac{283201}{276828}$, $\frac{190849}{10671882240}a^{31}+\frac{175571}{30236999680}a^{30}-\frac{15191}{1778647040}a^{29}-\frac{68713}{2667970560}a^{28}-\frac{456749}{3557294080}a^{27}-\frac{4000319}{90710999040}a^{26}-\frac{1692761}{5335941120}a^{25}+\frac{10374803}{45355499520}a^{24}-\frac{132187}{889323520}a^{23}+\frac{24149803}{22677749760}a^{22}+\frac{2342697}{1778647040}a^{21}+\frac{1197377}{889323520}a^{20}+\frac{14719147}{3557294080}a^{19}+\frac{116429737}{90710999040}a^{18}+\frac{32239853}{5335941120}a^{17}-\frac{323706359}{45355499520}a^{16}-\frac{9817319}{10671882240}a^{15}-\frac{2063220863}{90710999040}a^{14}-\frac{22028407}{889323520}a^{13}-\frac{221025369}{7559249920}a^{12}-\frac{421489}{8337408}a^{11}+\frac{421969}{16674816}a^{10}-\frac{4386557}{83374080}a^{9}+\frac{88936751}{708679680}a^{8}-\frac{934039}{20843520}a^{7}+\frac{41249447}{177169920}a^{6}+\frac{367367}{2605440}a^{5}+\frac{1753117}{11073120}a^{4}+\frac{1095461}{2605440}a^{3}+\frac{20647}{434240}a^{2}+\frac{119473}{217120}a-\frac{1000409}{1845520}$, $\frac{1611031}{181421998080}a^{31}+\frac{112417}{3943956480}a^{30}-\frac{158323}{5335941120}a^{29}-\frac{841}{115998720}a^{28}-\frac{7223393}{181421998080}a^{27}+\frac{198723}{1314652160}a^{26}-\frac{29233967}{90710999040}a^{25}-\frac{72783}{657326080}a^{24}-\frac{21860599}{45355499520}a^{23}-\frac{228391}{328663040}a^{22}+\frac{3334909}{5335941120}a^{21}-\frac{227557}{115998720}a^{20}+\frac{491417687}{181421998080}a^{19}-\frac{3568591}{3943956480}a^{18}+\frac{191721609}{30236999680}a^{17}-\frac{6220231}{1971978240}a^{16}+\frac{587556309}{60473999360}a^{15}+\frac{55981769}{3943956480}a^{14}+\frac{169827221}{45355499520}a^{13}+\frac{8891979}{328663040}a^{12}-\frac{254093}{3473920}a^{11}+\frac{23349}{1208320}a^{10}-\frac{48518053}{472453120}a^{9}-\frac{1344193}{30812160}a^{8}-\frac{17790507}{118113280}a^{7}-\frac{124357}{7703040}a^{6}+\frac{959969}{44292480}a^{5}-\frac{4629}{20060}a^{4}+\frac{369529}{868480}a^{3}-\frac{21731}{56640}a^{2}+\frac{17050253}{11073120}a+\frac{149201}{240720}$, $\frac{165211}{9071099904}a^{31}-\frac{402307}{30236999680}a^{30}+\frac{6795}{88932352}a^{29}-\frac{200431}{2667970560}a^{28}+\frac{529121}{3023699968}a^{27}+\frac{4606981}{30236999680}a^{26}-\frac{321643}{4535549952}a^{25}+\frac{23783381}{45355499520}a^{24}-\frac{1990307}{2267774976}a^{23}+\frac{6705503}{7559249920}a^{22}-\frac{503735}{266797056}a^{21}-\frac{186935}{533594112}a^{20}-\frac{224575}{153747456}a^{19}-\frac{226805401}{90710999040}a^{18}+\frac{24784439}{4535549952}a^{17}-\frac{152068459}{15118499840}a^{16}+\frac{184523843}{9071099904}a^{15}-\frac{642296561}{90710999040}a^{14}+\frac{81277685}{2267774976}a^{13}+\frac{110452653}{7559249920}a^{12}-\frac{1253}{8337408}a^{11}+\frac{4335961}{83374080}a^{10}-\frac{4950443}{70867968}a^{9}+\frac{56306707}{708679680}a^{8}-\frac{8122153}{35433984}a^{7}+\frac{7366677}{59056640}a^{6}-\frac{259101}{1476416}a^{5}-\frac{908723}{5536560}a^{4}+\frac{13139}{43424}a^{3}-\frac{267179}{434240}a^{2}+\frac{389917}{276828}a+\frac{760211}{5536560}$, $\frac{3671}{566943744}a^{31}+\frac{9721}{1889812480}a^{30}+\frac{679}{66699264}a^{29}-\frac{11353}{666992640}a^{28}-\frac{1777}{283471872}a^{27}-\frac{18577}{2834718720}a^{26}+\frac{9565}{377962496}a^{25}-\frac{213257}{11338874880}a^{24}-\frac{34073}{566943744}a^{23}+\frac{129729}{1889812480}a^{22}+\frac{1313}{4168704}a^{21}-\frac{277}{4168704}a^{20}-\frac{925}{5905664}a^{19}+\frac{540079}{472453120}a^{18}-\frac{2322505}{1133887488}a^{17}+\frac{25669159}{11338874880}a^{16}-\frac{368003}{141735936}a^{15}+\frac{1029899}{472453120}a^{14}+\frac{4495449}{377962496}a^{13}-\frac{297980101}{11338874880}a^{12}+\frac{37573}{2084352}a^{11}-\frac{689871}{27791360}a^{10}+\frac{1992167}{70867968}a^{9}+\frac{1381417}{354339840}a^{8}-\frac{822777}{11811328}a^{7}+\frac{3782267}{29528320}a^{6}-\frac{333607}{4429248}a^{5}+\frac{3894271}{22146240}a^{4}+\frac{3839}{32568}a^{3}-\frac{32351}{325680}a^{2}+\frac{151421}{553656}a-\frac{2581849}{2768280}$, $\frac{8485}{525860864}a^{31}-\frac{31}{3932160}a^{30}+\frac{10477}{231997440}a^{29}+\frac{119}{1966080}a^{28}+\frac{91309}{525860864}a^{27}+\frac{451}{1310720}a^{26}+\frac{745291}{1314652160}a^{25}+\frac{1123}{1966080}a^{24}+\frac{191317}{394395648}a^{23}+\frac{379}{983040}a^{22}-\frac{108829}{77332480}a^{21}-\frac{2741}{1966080}a^{20}-\frac{10622071}{2629304320}a^{19}-\frac{29263}{3932160}a^{18}-\frac{2644019}{262930432}a^{17}-\frac{25847}{1966080}a^{16}-\frac{78206813}{7887912960}a^{15}-\frac{20023}{3932160}a^{14}+\frac{5459063}{394395648}a^{13}+\frac{16169}{983040}a^{12}+\frac{167413}{2416640}a^{11}+\frac{4729}{61440}a^{10}+\frac{8516351}{61624320}a^{9}+\frac{7141}{30720}a^{8}+\frac{603401}{3081216}a^{7}+\frac{437}{2560}a^{6}-\frac{12929}{481440}a^{5}-\frac{59}{240}a^{4}-\frac{9961}{22656}a^{3}-\frac{79}{960}a^{2}-\frac{312757}{481440}a-\frac{29}{80}$, $\frac{150751}{22677749760}a^{31}-\frac{1562911}{90710999040}a^{30}+\frac{20881}{666992640}a^{29}-\frac{232433}{2667970560}a^{28}+\frac{1362647}{22677749760}a^{27}-\frac{6937949}{30236999680}a^{26}+\frac{572869}{11338874880}a^{25}-\frac{11360917}{45355499520}a^{24}-\frac{1955699}{5669437440}a^{23}+\frac{14413379}{22677749760}a^{22}-\frac{68399}{133398528}a^{21}+\frac{2631529}{889323520}a^{20}+\frac{2410351}{22677749760}a^{19}+\frac{221355099}{30236999680}a^{18}+\frac{11147407}{11338874880}a^{17}+\frac{7012067}{768737280}a^{16}+\frac{116549591}{22677749760}a^{15}-\frac{231389981}{30236999680}a^{14}+\frac{59870731}{5669437440}a^{13}-\frac{1349293331}{22677749760}a^{12}+\frac{361}{20355}a^{11}-\frac{10204523}{83374080}a^{10}+\frac{24662837}{708679680}a^{9}-\frac{84433589}{708679680}a^{8}+\frac{91193}{5536560}a^{7}+\frac{34400291}{177169920}a^{6}-\frac{3466837}{22146240}a^{5}+\frac{3760721}{5536560}a^{4}-\frac{120611}{651360}a^{3}+\frac{430791}{434240}a^{2}-\frac{153911}{1384140}a+\frac{5067413}{5536560}$ 1269343/90710999040a^31 + 85201/2667970560a^29 + 6679831/90710999040a^27 + 6902069/45355499520a^25 - 4480787/22677749760a^23 - 1808443/2667970560a^21 - 60743963/30236999680a^19 - 22762683/15118499840a^17 + 125489437/30236999680a^15 + 137011201/7559249920a^13 + 101003/5210880a^11 + 18115543/708679680a^9 + 1479937/177169920a^7 - 2666533/22146240a^5 - 148409/1302720a^3 + 383579/5536560a - 1, 2003171/181421998080a^31 + 102503/4535549952a^30 - 50143/5335941120a^29 + 1235/33349632a^28 + 2917209/60473999360a^27 + 505675/4535549952a^26 + 9704053/90710999040a^25 + 3725/47245312a^24 + 970367/15118499840a^23 - 52677/94490624a^22 - 1247551/5335941120a^21 - 96391/133398528a^20 - 17133853/181421998080a^19 - 1938871/1511849984a^18 - 154874873/90710999040a^17 + 1786465/1133887488a^16 + 234489/60473999360a^15 + 37028387/4535549952a^14 + 81595147/15118499840a^13 + 37462121/2267774976a^12 + 2984051/166748160a^11 + 579799/16674816a^10 + 19895587/472453120a^9 + 439169/17716992a^8 + 16633009/354339840a^7 - 162149/2214624a^6 - 3199763/22146240a^5 - 560113/4429248a^4 - 185713/2605440a^3 - 2531/21712a^2 - 4032907/11073120a + 32255/553656, 2294729/90710999040a^31 - 351443/18142199808a^30 + 63851/889323520a^29 - 38317/533594112a^28 + 2836371/30236999680a^27 + 372757/18142199808a^26 - 5851523/45355499520a^25 + 883861/3023699968a^24 - 16176931/22677749760a^23 + 3517255/4535549952a^22 - 3831109/2667970560a^21 + 122389/177864704a^20 - 10000207/90710999040a^19 - 2461091/18142199808a^18 + 163089283/45355499520a^17 - 22434353/3023699968a^16 + 1120934473/90710999040a^15 - 106884523/18142199808a^14 + 214184777/11338874880a^13 - 9589797/1511849984a^12 - 1571477/166748160a^11 + 200459/8337408a^10 - 14710463/354339840a^9 + 1715359/47245312a^8 - 16337059/177169920a^7 + 2797825/35433984a^6 - 238521/14764160a^5 - 156605/4429248a^4 + 15881/434240a^3 - 63875/260544a^2 + 883363/1384140a - 104625/369104, 29537/18142199808a^31 + 181449/6047399936a^30 - 29825/533594112a^29 + 12151/177864704a^28 - 538461/6047399936a^27 + 1457587/18142199808a^26 + 1342747/9071099904a^25 - 4294823/9071099904a^24 + 799843/4535549952a^23 - 2848303/4535549952a^22 + 385979/533594112a^21 - 345703/533594112a^20 + 5831633/18142199808a^19 + 1422649/6047399936a^18 + 4705921/9071099904a^17 + 61086043/9071099904a^16 - 99216343/18142199808a^15 + 241065331/18142199808a^14 - 14971195/4535549952a^13 + 48470135/4535549952a^12 - 14767/1042176a^11 - 44867/2779136a^10 - 3349079/141735936a^9 - 11462555/141735936a^8 + 363183/5905664a^7 - 4216685/35433984a^6 + 283717/4429248a^5 - 51685/4429248a^4 - 30407/260544a^3 + 222119/260544a^2 - 7095/369104a - 99111/369104, 23953/10671882240a^31 + 14385/3023699968a^30 + 10531/5335941120a^29 + 2635/266797056a^28 + 50147/3557294080a^27 - 392317/9071099904a^26 - 436273/5335941120a^25 - 58475/1511849984a^24 - 839257/2667970560a^23 - 258665/2267774976a^22 + 216847/1067188224a^21 - 108487/266797056a^20 + 9389233/10671882240a^19 - 968885/9071099904a^18 + 25982261/5335941120a^17 + 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10829/130272a^4 - 42811/1302720a^3 - 40829/260544a^2 - 35689/1107312a + 5611/65136, 773309/36284399616a^31 + 571361/45355499520a^30 + 124689/1778647040a^29 + 757/444661760a^28 + 1588085/36284399616a^27 - 7536823/45355499520a^26 - 17504497/90710999040a^25 - 12873061/22677749760a^24 - 2856005/9071099904a^23 - 8472229/11338874880a^22 - 3583237/5335941120a^21 - 120353/266797056a^20 + 55039259/60473999360a^19 + 128869201/45355499520a^18 + 38448379/6047399936a^17 + 99735979/7559249920a^16 + 1799627177/181421998080a^15 + 1046867081/45355499520a^14 - 38749685/9071099904a^13 - 26453193/3779624960a^12 - 75449/2826240a^11 - 1619519/27791360a^10 - 54574639/1417359360a^9 - 32345761/177169920a^8 - 1174967/23622656a^7 - 25824271/88584960a^6 + 3445487/22146240a^5 - 294477/7382080a^4 + 112479/173696a^3 + 603107/651360a^2 + 3139881/3691040a + 1969109/2768280, 10135/788791296a^31 + 5791/231997440a^30 - 4619/115998720a^29 - 1919/115998720a^28 - 102065/788791296a^27 - 13731/77332480a^26 + 48169/1971978240a^25 - 98923/115998720a^24 + 15127/65732608a^23 - 64099/57999360a^22 + 8929/115998720a^21 + 97301/115998720a^20 + 2994017/1314652160a^19 + 1403503/231997440a^18 + 264405/131465216a^17 + 1756847/115998720a^16 - 27530869/3943956480a^15 + 6992983/231997440a^14 - 2823017/197197824a^13 - 385709/57999360a^12 - 51361/7249920a^11 - 449599/3624960a^10 - 153557/5135360a^9 - 360091/1812480a^8 + 9823/192576a^7 - 49497/151040a^6 + 126723/641920a^5 - 2837/28320a^4 + 1529/11328a^3 + 56419/56640a^2 - 107971/240720a + 11009/4720, 969489/60473999360a^31 - 243583/45355499520a^30 + 206501/5335941120a^29 + 171/88932352a^28 - 14155861/181421998080a^27 - 3927551/45355499520a^26 - 5871597/30236999680a^25 + 1194701/4535549952a^24 - 2159141/15118499840a^23 + 3296579/3779624960a^22 - 2813119/5335941120a^21 + 511569/444661760a^20 + 190655011/181421998080a^19 - 1012315/9071099904a^18 + 640679819/90710999040a^17 + 26067959/22677749760a^16 + 1152736811/181421998080a^15 - 213358723/9071099904a^14 - 292752133/45355499520a^13 - 54309121/1417359360a^12 - 139339/10421760a^11 + 999/6947840a^10 - 23890749/472453120a^9 + 1169269/23622656a^8 - 10175369/118113280a^7 + 7374163/88584960a^6 + 2740717/44292480a^5 + 152233/276828a^4 + 1312559/2605440a^3 + 39383/217120a^2 + 892713/3691040a - 283201/276828, 190849/10671882240a^31 + 175571/30236999680a^30 - 15191/1778647040a^29 - 68713/2667970560a^28 - 456749/3557294080a^27 - 4000319/90710999040a^26 - 1692761/5335941120a^25 + 10374803/45355499520a^24 - 132187/889323520a^23 + 24149803/22677749760a^22 + 2342697/1778647040a^21 + 1197377/889323520a^20 + 14719147/3557294080a^19 + 116429737/90710999040a^18 + 32239853/5335941120a^17 - 323706359/45355499520a^16 - 9817319/10671882240a^15 - 2063220863/90710999040a^14 - 22028407/889323520a^13 - 221025369/7559249920a^12 - 421489/8337408a^11 + 421969/16674816a^10 - 4386557/83374080a^9 + 88936751/708679680a^8 - 934039/20843520a^7 + 41249447/177169920a^6 + 367367/2605440a^5 + 1753117/11073120a^4 + 1095461/2605440a^3 + 20647/434240a^2 + 119473/217120a - 1000409/1845520, 1611031/181421998080a^31 + 112417/3943956480a^30 - 158323/5335941120a^29 - 841/115998720a^28 - 7223393/181421998080a^27 + 198723/1314652160a^26 - 29233967/90710999040a^25 - 72783/657326080a^24 - 21860599/45355499520a^23 - 228391/328663040a^22 + 3334909/5335941120a^21 - 227557/115998720a^20 + 491417687/181421998080a^19 - 3568591/3943956480a^18 + 191721609/30236999680a^17 - 6220231/1971978240a^16 + 587556309/60473999360a^15 + 55981769/3943956480a^14 + 169827221/45355499520a^13 + 8891979/328663040a^12 - 254093/3473920a^11 + 23349/1208320a^10 - 48518053/472453120a^9 - 1344193/30812160a^8 - 17790507/118113280a^7 - 124357/7703040a^6 + 959969/44292480a^5 - 4629/20060a^4 + 369529/868480a^3 - 21731/56640a^2 + 17050253/11073120a + 149201/240720, 165211/9071099904a^31 - 402307/30236999680a^30 + 6795/88932352a^29 - 200431/2667970560a^28 + 529121/3023699968a^27 + 4606981/30236999680a^26 - 321643/4535549952a^25 + 23783381/45355499520a^24 - 1990307/2267774976a^23 + 6705503/7559249920a^22 - 503735/266797056a^21 - 186935/533594112a^20 - 224575/153747456a^19 - 226805401/90710999040a^18 + 24784439/4535549952a^17 - 152068459/15118499840a^16 + 184523843/9071099904a^15 - 642296561/90710999040a^14 + 81277685/2267774976a^13 + 110452653/7559249920a^12 - 1253/8337408a^11 + 4335961/83374080a^10 - 4950443/70867968a^9 + 56306707/708679680a^8 - 8122153/35433984a^7 + 7366677/59056640a^6 - 259101/1476416a^5 - 908723/5536560a^4 + 13139/43424a^3 - 267179/434240a^2 + 389917/276828a + 760211/5536560, 3671/566943744a^31 + 9721/1889812480a^30 + 679/66699264a^29 - 11353/666992640a^28 - 1777/283471872a^27 - 18577/2834718720a^26 + 9565/377962496a^25 - 213257/11338874880a^24 - 34073/566943744a^23 + 129729/1889812480a^22 + 1313/4168704a^21 - 277/4168704a^20 - 925/5905664a^19 + 540079/472453120a^18 - 2322505/1133887488a^17 + 25669159/11338874880a^16 - 368003/141735936a^15 + 1029899/472453120a^14 + 4495449/377962496a^13 - 297980101/11338874880a^12 + 37573/2084352a^11 - 689871/27791360a^10 + 1992167/70867968a^9 + 1381417/354339840a^8 - 822777/11811328a^7 + 3782267/29528320a^6 - 333607/4429248a^5 + 3894271/22146240a^4 + 3839/32568a^3 - 32351/325680a^2 + 151421/553656a - 2581849/2768280, 8485/525860864a^31 - 31/3932160a^30 + 10477/231997440a^29 + 119/1966080a^28 + 91309/525860864a^27 + 451/1310720a^26 + 745291/1314652160a^25 + 1123/1966080a^24 + 191317/394395648a^23 + 379/983040a^22 - 108829/77332480a^21 - 2741/1966080a^20 - 10622071/2629304320a^19 - 29263/3932160a^18 - 2644019/262930432a^17 - 25847/1966080a^16 - 78206813/7887912960a^15 - 20023/3932160a^14 + 5459063/394395648a^13 + 16169/983040a^12 + 167413/2416640a^11 + 4729/61440a^10 + 8516351/61624320a^9 + 7141/30720a^8 + 603401/3081216a^7 + 437/2560a^6 - 12929/481440a^5 - 59/240a^4 - 9961/22656a^3 - 79/960a^2 - 312757/481440a - 29/80, 150751/22677749760a^31 - 1562911/90710999040a^30 + 20881/666992640a^29 - 232433/2667970560a^28 + 1362647/22677749760a^27 - 6937949/30236999680a^26 + 572869/11338874880a^25 - 11360917/45355499520a^24 - 1955699/5669437440a^23 + 14413379/22677749760a^22 - 68399/133398528a^21 + 2631529/889323520a^20 + 2410351/22677749760a^19 + 221355099/30236999680a^18 + 11147407/11338874880a^17 + 7012067/768737280a^16 + 116549591/22677749760a^15 - 231389981/30236999680a^14 + 59870731/5669437440a^13 - 1349293331/22677749760a^12 + 361/20355a^11 - 10204523/83374080a^10 + 24662837/708679680a^9 - 84433589/708679680a^8 + 91193/5536560a^7 + 34400291/177169920a^6 - 3466837/22146240a^5 + 3760721/5536560a^4 - 120611/651360a^3 + 430791/434240a^2 - 153911/1384140a + 5067413/5536560 (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:	$ 495183335125.27386 $ (assuming GRH)	sage: K.regulator() gp: K.reg magma: Regulator(K); oscar: regulator(K)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{16}\cdot 495183335125.27386 \cdot 12}{12\cdot\sqrt{126945864674316252168838917102904812361796913463296}}\cr\approx \mathstrut & 0.259319177364052 \end{aligned}\] (assuming GRH)

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

x = polygen(QQ); K.<a> = NumberField(x^32 + 2*x^30 + x^28 - 6*x^26 - 28*x^24 - 58*x^22 + 9*x^20 + 230*x^18 + 593*x^16 + 920*x^14 + 144*x^12 - 3712*x^10 - 7168*x^8 - 6144*x^6 + 4096*x^4 + 32768*x^2 + 65536)

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 + 2*x^30 + x^28 - 6*x^26 - 28*x^24 - 58*x^22 + 9*x^20 + 230*x^18 + 593*x^16 + 920*x^14 + 144*x^12 - 3712*x^10 - 7168*x^8 - 6144*x^6 + 4096*x^4 + 32768*x^2 + 65536, 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 + 2*x^30 + x^28 - 6*x^26 - 28*x^24 - 58*x^22 + 9*x^20 + 230*x^18 + 593*x^16 + 920*x^14 + 144*x^12 - 3712*x^10 - 7168*x^8 - 6144*x^6 + 4096*x^4 + 32768*x^2 + 65536);

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 + 2*x^30 + x^28 - 6*x^26 - 28*x^24 - 58*x^22 + 9*x^20 + 230*x^18 + 593*x^16 + 920*x^14 + 144*x^12 - 3712*x^10 - 7168*x^8 - 6144*x^6 + 4096*x^4 + 32768*x^2 + 65536);

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

$C_2^6:S_4$ (as 32T96908):

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 1536

The 80 conjugacy class representatives for $C_2^6:S_4$

Character table for $C_2^6:S_4$

Intermediate fields

$\Q(\sqrt{3}) $, $\Q(\sqrt{-3}) $, $\Q(\sqrt{-1}) $, 4.4.52176.1, $\Q(\zeta_{12})$, 8.0.3356639025408.1, 8.0.372959891712.1, 8.8.372959891712.1, 8.8.3356639025408.1, 8.8.24501014784.1, 8.0.24501014784.3, 8.0.2722334976.1, 16.0.600299725445786566656.1, 16.0.11267025546891968069566464.4, 16.16.11267025546891968069566464.1, 16.0.11267025546891968069566464.1, 16.0.11267025546891968069566464.2, 16.0.139099080825826766290944.1, 16.0.11267025546891968069566464.3

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:	not computed

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	${\href{/padicField/5.6.0.1}{6} }^{4}{,}\,{\href{/padicField/5.2.0.1}{2} }^{4}$	${\href{/padicField/7.4.0.1}{4} }^{8}$	${\href{/padicField/11.6.0.1}{6} }^{4}{,}\,{\href{/padicField/11.2.0.1}{2} }^{4}$	${\href{/padicField/13.8.0.1}{8} }^{4}$	${\href{/padicField/17.4.0.1}{4} }^{4}{,}\,{\href{/padicField/17.2.0.1}{2} }^{8}$	${\href{/padicField/19.4.0.1}{4} }^{8}$	${\href{/padicField/23.6.0.1}{6} }^{4}{,}\,{\href{/padicField/23.2.0.1}{2} }^{4}$	${\href{/padicField/29.6.0.1}{6} }^{4}{,}\,{\href{/padicField/29.2.0.1}{2} }^{4}$	${\href{/padicField/31.2.0.1}{2} }^{16}$	${\href{/padicField/37.2.0.1}{2} }^{16}$	${\href{/padicField/41.8.0.1}{8} }^{4}$	${\href{/padicField/43.6.0.1}{6} }^{4}{,}\,{\href{/padicField/43.2.0.1}{2} }^{4}$	${\href{/padicField/47.6.0.1}{6} }^{4}{,}\,{\href{/padicField/47.2.0.1}{2} }^{4}$	${\href{/padicField/53.6.0.1}{6} }^{4}{,}\,{\href{/padicField/53.2.0.1}{2} }^{4}$	${\href{/padicField/59.2.0.1}{2} }^{16}$

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	2.8.8.1	$x^{8} + 8 x^{7} + 32 x^{6} + 82 x^{5} + 148 x^{4} + 184 x^{3} + 137 x^{2} + 44 x + 5$	$2$	$4$	$8$	$C_4\times C_2$	$[2]^{4}$
	2.8.8.1	$x^{8} + 8 x^{7} + 32 x^{6} + 82 x^{5} + 148 x^{4} + 184 x^{3} + 137 x^{2} + 44 x + 5$	$2$	$4$	$8$	$C_4\times C_2$	$[2]^{4}$
	2.8.8.1	$x^{8} + 8 x^{7} + 32 x^{6} + 82 x^{5} + 148 x^{4} + 184 x^{3} + 137 x^{2} + 44 x + 5$	$2$	$4$	$8$	$C_4\times C_2$	$[2]^{4}$
	2.8.8.1	$x^{8} + 8 x^{7} + 32 x^{6} + 82 x^{5} + 148 x^{4} + 184 x^{3} + 137 x^{2} + 44 x + 5$	$2$	$4$	$8$	$C_4\times C_2$	$[2]^{4}$
$3$ 3	3.4.2.1	$x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$
	3.4.2.1	$x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$
	3.4.2.1	$x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$
	3.4.2.1	$x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$
	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	$x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$	$2$	$4$	$4$	$C_4\times C_2$	$[\ ]_{2}^{4}$
$137$ 137	137.2.0.1	$x^{2} + 131 x + 3$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
	137.2.0.1	$x^{2} + 131 x + 3$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
	137.2.0.1	$x^{2} + 131 x + 3$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
	137.2.0.1	$x^{2} + 131 x + 3$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
	137.4.2.1	$x^{4} + 20538 x^{3} + 106780719 x^{2} + 13640908302 x + 496637065$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$
	137.4.0.1	$x^{4} + x^{2} + 95 x + 3$	$1$	$4$	$0$	$C_4$	$[\ ]^{4}$
	137.4.0.1	$x^{4} + x^{2} + 95 x + 3$	$1$	$4$	$0$	$C_4$	$[\ ]^{4}$
	137.4.0.1	$x^{4} + x^{2} + 95 x + 3$	$1$	$4$	$0$	$C_4$	$[\ ]^{4}$
	137.4.0.1	$x^{4} + x^{2} + 95 x + 3$	$1$	$4$	$0$	$C_4$	$[\ ]^{4}$
	137.4.2.1	$x^{4} + 20538 x^{3} + 106780719 x^{2} + 13640908302 x + 496637065$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$
$1087$ 1087		Deg $2$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
		Deg $2$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
		Deg $2$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
		Deg $2$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
		Deg $2$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
		Deg $2$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
		Deg $2$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
		Deg $2$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
		Deg $4$	$2$	$2$	$2$
		Deg $4$	$2$	$2$	$2$
		Deg $4$	$2$	$2$	$2$
		Deg $4$	$2$	$2$	$2$

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