LMFDB - Number field 32.0.42655280577413385127292283461599409580230712890625.1

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^32 - 2*x^31 - 14*x^30 + 32*x^29 + 110*x^28 - 256*x^27 - 822*x^26 + 1761*x^25 + 5076*x^24 - 6234*x^23 - 33617*x^22 + 7049*x^21 + 145847*x^20 + 80587*x^19 - 200297*x^18 - 1040496*x^17 + 909785*x^16 - 126545*x^15 + 4807173*x^14 - 6468687*x^13 + 4092091*x^12 - 12326083*x^11 + 15573178*x^10 - 9966741*x^9 + 15430857*x^8 - 15762104*x^7 + 7858456*x^6 - 7358464*x^5 + 5444944*x^4 - 823424*x^3 + 88384*x^2 - 5760*x + 256)

gp: K = bnfinit(y^32 - 2*y^31 - 14*y^30 + 32*y^29 + 110*y^28 - 256*y^27 - 822*y^26 + 1761*y^25 + 5076*y^24 - 6234*y^23 - 33617*y^22 + 7049*y^21 + 145847*y^20 + 80587*y^19 - 200297*y^18 - 1040496*y^17 + 909785*y^16 - 126545*y^15 + 4807173*y^14 - 6468687*y^13 + 4092091*y^12 - 12326083*y^11 + 15573178*y^10 - 9966741*y^9 + 15430857*y^8 - 15762104*y^7 + 7858456*y^6 - 7358464*y^5 + 5444944*y^4 - 823424*y^3 + 88384*y^2 - 5760*y + 256, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^32 - 2*x^31 - 14*x^30 + 32*x^29 + 110*x^28 - 256*x^27 - 822*x^26 + 1761*x^25 + 5076*x^24 - 6234*x^23 - 33617*x^22 + 7049*x^21 + 145847*x^20 + 80587*x^19 - 200297*x^18 - 1040496*x^17 + 909785*x^16 - 126545*x^15 + 4807173*x^14 - 6468687*x^13 + 4092091*x^12 - 12326083*x^11 + 15573178*x^10 - 9966741*x^9 + 15430857*x^8 - 15762104*x^7 + 7858456*x^6 - 7358464*x^5 + 5444944*x^4 - 823424*x^3 + 88384*x^2 - 5760*x + 256);

oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^32 - 2*x^31 - 14*x^30 + 32*x^29 + 110*x^28 - 256*x^27 - 822*x^26 + 1761*x^25 + 5076*x^24 - 6234*x^23 - 33617*x^22 + 7049*x^21 + 145847*x^20 + 80587*x^19 - 200297*x^18 - 1040496*x^17 + 909785*x^16 - 126545*x^15 + 4807173*x^14 - 6468687*x^13 + 4092091*x^12 - 12326083*x^11 + 15573178*x^10 - 9966741*x^9 + 15430857*x^8 - 15762104*x^7 + 7858456*x^6 - 7358464*x^5 + 5444944*x^4 - 823424*x^3 + 88384*x^2 - 5760*x + 256)

$ x^{32} - 2 x^{31} - 14 x^{30} + 32 x^{29} + 110 x^{28} - 256 x^{27} - 822 x^{26} + 1761 x^{25} + \cdots + 256 $ x^32 - 2*x^31 - 14*x^30 + 32*x^29 + 110*x^28 - 256*x^27 - 822*x^26 + 1761*x^25 + 5076*x^24 - 6234*x^23 - 33617*x^22 + 7049*x^21 + 145847*x^20 + 80587*x^19 - 200297*x^18 - 1040496*x^17 + 909785*x^16 - 126545*x^15 + 4807173*x^14 - 6468687*x^13 + 4092091*x^12 - 12326083*x^11 + 15573178*x^10 - 9966741*x^9 + 15430857*x^8 - 15762104*x^7 + 7858456*x^6 - 7358464*x^5 + 5444944*x^4 - 823424*x^3 + 88384*x^2 - 5760*x + 256

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:	$42655280577413385127292283461599409580230712890625$ 42655280577413385127292283461599409580230712890625 $\medspace = 3^{16}\cdot 5^{24}\cdot 7^{16}\cdot 29^{8}$	sage: K.disc() gp: K.disc magma: OK := Integers(K); Discriminant(OK); oscar: OK = ring_of_integers(K); discriminant(OK)
Root discriminant:	$35.56$	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:	$3^{1/2}5^{3/4}7^{1/2}29^{1/2}\approx 82.51561665216384$
Ramified primes:	$3$, $5$, $7$, $29$ 3, 5, 7, 29	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 $a$)

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{2}a^{15}-\frac{1}{2}a^{14}-\frac{1}{2}a^{13}-\frac{1}{2}a^{12}-\frac{1}{2}a^{11}-\frac{1}{2}a^{10}-\frac{1}{2}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{2}a^{16}-\frac{1}{2}a$, $\frac{1}{2}a^{17}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{18}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{19}-\frac{1}{2}a^{4}$, $\frac{1}{10}a^{20}+\frac{1}{5}a^{19}+\frac{1}{10}a^{17}+\frac{1}{5}a^{16}-\frac{1}{5}a^{15}+\frac{2}{5}a^{14}-\frac{2}{5}a^{13}-\frac{1}{5}a^{12}-\frac{2}{5}a^{11}+\frac{2}{5}a^{10}+\frac{1}{5}a^{9}+\frac{2}{5}a^{8}-\frac{2}{5}a^{7}-\frac{1}{5}a^{6}-\frac{3}{10}a^{5}-\frac{2}{5}a^{4}+\frac{1}{5}a^{3}-\frac{1}{2}a^{2}-\frac{1}{5}a-\frac{2}{5}$, $\frac{1}{10}a^{21}+\frac{1}{10}a^{19}+\frac{1}{10}a^{18}-\frac{1}{10}a^{16}-\frac{1}{5}a^{15}-\frac{1}{5}a^{14}-\frac{2}{5}a^{13}+\frac{1}{5}a^{11}+\frac{2}{5}a^{10}-\frac{1}{5}a^{8}-\frac{2}{5}a^{7}+\frac{1}{10}a^{6}+\frac{1}{5}a^{5}-\frac{1}{2}a^{4}+\frac{1}{10}a^{3}-\frac{1}{5}a^{2}-\frac{1}{2}a-\frac{1}{5}$, $\frac{1}{20}a^{22}-\frac{1}{20}a^{20}+\frac{1}{10}a^{19}+\frac{1}{10}a^{17}-\frac{1}{20}a^{16}-\frac{3}{20}a^{15}-\frac{7}{20}a^{14}-\frac{7}{20}a^{13}+\frac{1}{20}a^{12}-\frac{3}{20}a^{11}+\frac{7}{20}a^{10}-\frac{1}{20}a^{9}-\frac{7}{20}a^{8}-\frac{3}{10}a^{7}+\frac{1}{20}a^{6}+\frac{3}{10}a^{5}-\frac{1}{20}a^{4}+\frac{9}{20}a^{3}+\frac{1}{4}a^{2}-\frac{2}{5}a+\frac{2}{5}$, $\frac{1}{20}a^{23}-\frac{1}{20}a^{21}-\frac{1}{5}a^{19}+\frac{1}{10}a^{18}-\frac{3}{20}a^{17}+\frac{3}{20}a^{16}-\frac{3}{20}a^{15}+\frac{1}{4}a^{14}+\frac{9}{20}a^{13}+\frac{1}{20}a^{12}-\frac{1}{4}a^{11}-\frac{9}{20}a^{10}+\frac{9}{20}a^{9}+\frac{3}{10}a^{8}+\frac{9}{20}a^{7}-\frac{1}{2}a^{6}+\frac{1}{4}a^{5}-\frac{3}{20}a^{4}+\frac{1}{20}a^{3}+\frac{1}{10}a^{2}+\frac{1}{10}a+\frac{2}{5}$, $\frac{1}{20}a^{24}-\frac{1}{20}a^{20}+\frac{1}{10}a^{19}-\frac{3}{20}a^{18}-\frac{1}{20}a^{17}+\frac{1}{5}a^{16}+\frac{1}{5}a^{15}+\frac{2}{5}a^{14}+\frac{2}{5}a^{13}-\frac{1}{10}a^{12}+\frac{1}{10}a^{11}+\frac{1}{10}a^{10}+\frac{3}{20}a^{9}+\frac{2}{5}a^{8}-\frac{1}{10}a^{7}+\frac{2}{5}a^{6}+\frac{1}{20}a^{5}+\frac{1}{5}a^{4}+\frac{9}{20}a^{3}+\frac{7}{20}a^{2}+\frac{1}{10}a-\frac{2}{5}$, $\frac{1}{40}a^{25}-\frac{1}{40}a^{24}-\frac{1}{40}a^{22}-\frac{1}{40}a^{21}+\frac{1}{8}a^{19}-\frac{1}{5}a^{18}-\frac{1}{40}a^{17}+\frac{3}{40}a^{16}-\frac{1}{8}a^{15}+\frac{11}{40}a^{14}-\frac{7}{40}a^{13}+\frac{11}{40}a^{12}+\frac{19}{40}a^{11}-\frac{1}{20}a^{10}-\frac{1}{20}a^{9}+\frac{1}{40}a^{8}+\frac{3}{10}a^{7}-\frac{11}{40}a^{5}+\frac{1}{20}a^{4}-\frac{9}{40}a^{3}-\frac{1}{4}a^{2}-\frac{1}{10}a+\frac{2}{5}$, $\frac{1}{440}a^{26}+\frac{1}{220}a^{25}-\frac{3}{440}a^{24}-\frac{3}{440}a^{23}-\frac{1}{44}a^{22}-\frac{21}{440}a^{21}-\frac{21}{440}a^{20}-\frac{1}{440}a^{19}+\frac{31}{440}a^{18}+\frac{1}{220}a^{17}-\frac{2}{11}a^{16}+\frac{1}{10}a^{15}+\frac{9}{44}a^{14}-\frac{109}{220}a^{13}-\frac{23}{110}a^{12}+\frac{131}{440}a^{11}-\frac{4}{11}a^{10}-\frac{41}{440}a^{9}-\frac{163}{440}a^{8}+\frac{71}{220}a^{7}-\frac{153}{440}a^{6}-\frac{21}{440}a^{5}-\frac{163}{440}a^{4}-\frac{137}{440}a^{3}-\frac{47}{110}a^{2}+\frac{13}{110}a+\frac{14}{55}$, $\frac{1}{440}a^{27}+\frac{1}{110}a^{25}-\frac{1}{55}a^{24}-\frac{1}{110}a^{23}+\frac{1}{44}a^{22}+\frac{1}{44}a^{21}+\frac{19}{440}a^{20}-\frac{1}{5}a^{19}+\frac{9}{55}a^{18}-\frac{51}{440}a^{17}-\frac{1}{88}a^{16}+\frac{101}{440}a^{15}-\frac{211}{440}a^{14}-\frac{107}{440}a^{13}-\frac{101}{220}a^{12}-\frac{59}{440}a^{11}+\frac{191}{440}a^{10}+\frac{19}{88}a^{9}+\frac{21}{88}a^{8}-\frac{217}{440}a^{7}+\frac{87}{440}a^{6}+\frac{1}{4}a^{5}+\frac{189}{440}a^{4}+\frac{37}{88}a^{3}+\frac{26}{55}a^{2}-\frac{53}{110}a+\frac{16}{55}$, $\frac{1}{880}a^{28}+\frac{3}{440}a^{25}-\frac{7}{440}a^{24}-\frac{1}{40}a^{23}-\frac{1}{55}a^{22}-\frac{7}{880}a^{21}-\frac{1}{220}a^{20}-\frac{83}{440}a^{19}+\frac{89}{880}a^{18}+\frac{37}{176}a^{17}-\frac{217}{880}a^{16}+\frac{119}{880}a^{15}+\frac{391}{880}a^{14}-\frac{47}{220}a^{13}-\frac{329}{880}a^{12}-\frac{91}{880}a^{11}+\frac{15}{176}a^{10}+\frac{45}{176}a^{9}+\frac{149}{880}a^{8}-\frac{17}{176}a^{7}+\frac{97}{440}a^{6}+\frac{59}{176}a^{5}-\frac{87}{880}a^{4}-\frac{161}{440}a^{3}-\frac{41}{220}a^{2}+\frac{23}{110}a+\frac{1}{11}$, $\frac{1}{1760}a^{29}-\frac{1}{880}a^{27}+\frac{1}{176}a^{25}+\frac{3}{440}a^{24}-\frac{17}{880}a^{23}+\frac{3}{160}a^{22}+\frac{7}{880}a^{21}-\frac{39}{880}a^{20}+\frac{51}{1760}a^{19}-\frac{73}{352}a^{18}+\frac{93}{1760}a^{17}-\frac{63}{352}a^{16}-\frac{119}{1760}a^{15}-\frac{35}{176}a^{14}-\frac{347}{1760}a^{13}-\frac{135}{352}a^{12}+\frac{771}{1760}a^{11}-\frac{23}{160}a^{10}+\frac{173}{352}a^{9}-\frac{145}{352}a^{8}-\frac{111}{440}a^{7}+\frac{731}{1760}a^{6}-\frac{137}{1760}a^{5}-\frac{81}{880}a^{4}+\frac{9}{55}a^{3}-\frac{31}{220}a^{2}+\frac{23}{110}a+\frac{4}{55}$, $\frac{1}{78\!\cdots\!80}a^{30}-\frac{273647326495943}{96\!\cdots\!40}a^{29}+\frac{20\!\cdots\!47}{35\!\cdots\!40}a^{28}-\frac{76\!\cdots\!67}{98\!\cdots\!60}a^{27}-\frac{18\!\cdots\!09}{39\!\cdots\!40}a^{26}-\frac{10\!\cdots\!61}{98\!\cdots\!60}a^{25}+\frac{78\!\cdots\!25}{78\!\cdots\!08}a^{24}+\frac{22\!\cdots\!41}{15\!\cdots\!16}a^{23}+\frac{22\!\cdots\!97}{98\!\cdots\!60}a^{22}+\frac{12\!\cdots\!87}{39\!\cdots\!40}a^{21}-\frac{14\!\cdots\!29}{78\!\cdots\!80}a^{20}-\frac{16\!\cdots\!97}{17\!\cdots\!80}a^{19}+\frac{15\!\cdots\!59}{78\!\cdots\!80}a^{18}-\frac{21\!\cdots\!61}{78\!\cdots\!80}a^{17}-\frac{16\!\cdots\!97}{78\!\cdots\!80}a^{16}+\frac{17\!\cdots\!03}{19\!\cdots\!20}a^{15}+\frac{14\!\cdots\!21}{78\!\cdots\!80}a^{14}+\frac{31\!\cdots\!67}{15\!\cdots\!16}a^{13}+\frac{17\!\cdots\!21}{78\!\cdots\!80}a^{12}+\frac{17\!\cdots\!59}{14\!\cdots\!56}a^{11}-\frac{58\!\cdots\!49}{19\!\cdots\!80}a^{10}-\frac{30\!\cdots\!07}{78\!\cdots\!80}a^{9}-\frac{48\!\cdots\!29}{39\!\cdots\!40}a^{8}-\frac{16\!\cdots\!73}{78\!\cdots\!80}a^{7}-\frac{18\!\cdots\!59}{78\!\cdots\!80}a^{6}+\frac{88\!\cdots\!51}{19\!\cdots\!20}a^{5}+\frac{36\!\cdots\!09}{98\!\cdots\!60}a^{4}-\frac{11\!\cdots\!29}{24\!\cdots\!90}a^{3}-\frac{19\!\cdots\!79}{49\!\cdots\!80}a^{2}-\frac{42\!\cdots\!94}{12\!\cdots\!95}a+\frac{52\!\cdots\!42}{12\!\cdots\!95}$, $\frac{1}{28\!\cdots\!40}a^{31}-\frac{20\!\cdots\!61}{72\!\cdots\!60}a^{30}+\frac{30\!\cdots\!61}{13\!\cdots\!20}a^{29}-\frac{17\!\cdots\!51}{72\!\cdots\!60}a^{28}+\frac{13\!\cdots\!83}{14\!\cdots\!20}a^{27}-\frac{49\!\cdots\!67}{72\!\cdots\!60}a^{26}-\frac{14\!\cdots\!09}{13\!\cdots\!20}a^{25}+\frac{59\!\cdots\!09}{28\!\cdots\!40}a^{24}-\frac{20\!\cdots\!51}{28\!\cdots\!24}a^{23}+\frac{60\!\cdots\!27}{14\!\cdots\!20}a^{22}+\frac{10\!\cdots\!43}{57\!\cdots\!48}a^{21}+\frac{98\!\cdots\!79}{28\!\cdots\!40}a^{20}+\frac{36\!\cdots\!13}{28\!\cdots\!40}a^{19}-\frac{24\!\cdots\!15}{14\!\cdots\!28}a^{18}-\frac{12\!\cdots\!31}{70\!\cdots\!40}a^{17}-\frac{17\!\cdots\!99}{14\!\cdots\!20}a^{16}+\frac{10\!\cdots\!89}{28\!\cdots\!40}a^{15}+\frac{22\!\cdots\!53}{57\!\cdots\!48}a^{14}-\frac{60\!\cdots\!01}{28\!\cdots\!40}a^{13}-\frac{36\!\cdots\!41}{28\!\cdots\!40}a^{12}+\frac{10\!\cdots\!49}{28\!\cdots\!40}a^{11}-\frac{83\!\cdots\!61}{28\!\cdots\!40}a^{10}-\frac{11\!\cdots\!03}{36\!\cdots\!80}a^{9}-\frac{12\!\cdots\!97}{28\!\cdots\!40}a^{8}-\frac{13\!\cdots\!53}{28\!\cdots\!40}a^{7}+\frac{33\!\cdots\!19}{14\!\cdots\!20}a^{6}-\frac{30\!\cdots\!07}{90\!\cdots\!20}a^{5}+\frac{10\!\cdots\!49}{72\!\cdots\!56}a^{4}-\frac{41\!\cdots\!89}{90\!\cdots\!20}a^{3}-\frac{46\!\cdots\!69}{90\!\cdots\!20}a^{2}+\frac{37\!\cdots\!53}{22\!\cdots\!55}a-\frac{90\!\cdots\!38}{22\!\cdots\!55}$ 1, a, a^2, a^3, a^4, a^5, a^6, a^7, a^8, a^9, a^10, a^11, a^12, a^13, a^14, 1/2*a^15 - 1/2*a^14 - 1/2*a^13 - 1/2*a^12 - 1/2*a^11 - 1/2*a^10 - 1/2*a^9 - 1/2*a^8 - 1/2*a^7 - 1/2*a^6 - 1/2*a^5 - 1/2*a^4 - 1/2*a^3 - 1/2*a^2 - 1/2*a, 1/2*a^16 - 1/2*a, 1/2*a^17 - 1/2*a^2, 1/2*a^18 - 1/2*a^3, 1/2*a^19 - 1/2*a^4, 1/10*a^20 + 1/5*a^19 + 1/10*a^17 + 1/5*a^16 - 1/5*a^15 + 2/5*a^14 - 2/5*a^13 - 1/5*a^12 - 2/5*a^11 + 2/5*a^10 + 1/5*a^9 + 2/5*a^8 - 2/5*a^7 - 1/5*a^6 - 3/10*a^5 - 2/5*a^4 + 1/5*a^3 - 1/2*a^2 - 1/5*a - 2/5, 1/10*a^21 + 1/10*a^19 + 1/10*a^18 - 1/10*a^16 - 1/5*a^15 - 1/5*a^14 - 2/5*a^13 + 1/5*a^11 + 2/5*a^10 - 1/5*a^8 - 2/5*a^7 + 1/10*a^6 + 1/5*a^5 - 1/2*a^4 + 1/10*a^3 - 1/5*a^2 - 1/2*a - 1/5, 1/20*a^22 - 1/20*a^20 + 1/10*a^19 + 1/10*a^17 - 1/20*a^16 - 3/20*a^15 - 7/20*a^14 - 7/20*a^13 + 1/20*a^12 - 3/20*a^11 + 7/20*a^10 - 1/20*a^9 - 7/20*a^8 - 3/10*a^7 + 1/20*a^6 + 3/10*a^5 - 1/20*a^4 + 9/20*a^3 + 1/4*a^2 - 2/5*a + 2/5, 1/20*a^23 - 1/20*a^21 - 1/5*a^19 + 1/10*a^18 - 3/20*a^17 + 3/20*a^16 - 3/20*a^15 + 1/4*a^14 + 9/20*a^13 + 1/20*a^12 - 1/4*a^11 - 9/20*a^10 + 9/20*a^9 + 3/10*a^8 + 9/20*a^7 - 1/2*a^6 + 1/4*a^5 - 3/20*a^4 + 1/20*a^3 + 1/10*a^2 + 1/10*a + 2/5, 1/20*a^24 - 1/20*a^20 + 1/10*a^19 - 3/20*a^18 - 1/20*a^17 + 1/5*a^16 + 1/5*a^15 + 2/5*a^14 + 2/5*a^13 - 1/10*a^12 + 1/10*a^11 + 1/10*a^10 + 3/20*a^9 + 2/5*a^8 - 1/10*a^7 + 2/5*a^6 + 1/20*a^5 + 1/5*a^4 + 9/20*a^3 + 7/20*a^2 + 1/10*a - 2/5, 1/40*a^25 - 1/40*a^24 - 1/40*a^22 - 1/40*a^21 + 1/8*a^19 - 1/5*a^18 - 1/40*a^17 + 3/40*a^16 - 1/8*a^15 + 11/40*a^14 - 7/40*a^13 + 11/40*a^12 + 19/40*a^11 - 1/20*a^10 - 1/20*a^9 + 1/40*a^8 + 3/10*a^7 - 11/40*a^5 + 1/20*a^4 - 9/40*a^3 - 1/4*a^2 - 1/10*a + 2/5, 1/440*a^26 + 1/220*a^25 - 3/440*a^24 - 3/440*a^23 - 1/44*a^22 - 21/440*a^21 - 21/440*a^20 - 1/440*a^19 + 31/440*a^18 + 1/220*a^17 - 2/11*a^16 + 1/10*a^15 + 9/44*a^14 - 109/220*a^13 - 23/110*a^12 + 131/440*a^11 - 4/11*a^10 - 41/440*a^9 - 163/440*a^8 + 71/220*a^7 - 153/440*a^6 - 21/440*a^5 - 163/440*a^4 - 137/440*a^3 - 47/110*a^2 + 13/110*a + 14/55, 1/440*a^27 + 1/110*a^25 - 1/55*a^24 - 1/110*a^23 + 1/44*a^22 + 1/44*a^21 + 19/440*a^20 - 1/5*a^19 + 9/55*a^18 - 51/440*a^17 - 1/88*a^16 + 101/440*a^15 - 211/440*a^14 - 107/440*a^13 - 101/220*a^12 - 59/440*a^11 + 191/440*a^10 + 19/88*a^9 + 21/88*a^8 - 217/440*a^7 + 87/440*a^6 + 1/4*a^5 + 189/440*a^4 + 37/88*a^3 + 26/55*a^2 - 53/110*a + 16/55, 1/880*a^28 + 3/440*a^25 - 7/440*a^24 - 1/40*a^23 - 1/55*a^22 - 7/880*a^21 - 1/220*a^20 - 83/440*a^19 + 89/880*a^18 + 37/176*a^17 - 217/880*a^16 + 119/880*a^15 + 391/880*a^14 - 47/220*a^13 - 329/880*a^12 - 91/880*a^11 + 15/176*a^10 + 45/176*a^9 + 149/880*a^8 - 17/176*a^7 + 97/440*a^6 + 59/176*a^5 - 87/880*a^4 - 161/440*a^3 - 41/220*a^2 + 23/110*a + 1/11, 1/1760*a^29 - 1/880*a^27 + 1/176*a^25 + 3/440*a^24 - 17/880*a^23 + 3/160*a^22 + 7/880*a^21 - 39/880*a^20 + 51/1760*a^19 - 73/352*a^18 + 93/1760*a^17 - 63/352*a^16 - 119/1760*a^15 - 35/176*a^14 - 347/1760*a^13 - 135/352*a^12 + 771/1760*a^11 - 23/160*a^10 + 173/352*a^9 - 145/352*a^8 - 111/440*a^7 + 731/1760*a^6 - 137/1760*a^5 - 81/880*a^4 + 9/55*a^3 - 31/220*a^2 + 23/110*a + 4/55, 1/788329307332495206080*a^30 - 273647326495943/9613772040640185440*a^29 + 2039916433878847/35833150333295236640*a^28 - 76118053460467967/98541163416561900760*a^27 - 188096218806253209/394164653666247603040*a^26 - 1004532165772086261/98541163416561900760*a^25 + 786067483935272425/78832930733249520608*a^24 + 2255536336218377941/157665861466499041216*a^23 + 2230243329975676297/98541163416561900760*a^22 + 12468895811385734187/394164653666247603040*a^21 - 14883716100705914329/788329307332495206080*a^20 - 168852905661253897/1747958552843670080*a^19 + 158686810906565794859/788329307332495206080*a^18 - 21787505725541715561/788329307332495206080*a^17 - 169817880117851536397/788329307332495206080*a^16 + 17957623019501153303/197082326833123801520*a^15 + 148188667037518430921/788329307332495206080*a^14 + 31496076079829491367/157665861466499041216*a^13 + 178555436550903161521/788329307332495206080*a^12 + 1798125515017844659/14333260133318094656*a^11 - 5864877093093421449/19227544081280370880*a^10 - 309177294374850626007/788329307332495206080*a^9 - 48212931678796372929/394164653666247603040*a^8 - 167162946317890604973/788329307332495206080*a^7 - 183105746058106056059/788329307332495206080*a^6 + 88070177429046161951/197082326833123801520*a^5 + 36528014430284199709/98541163416561900760*a^4 - 11690413197754584429/24635290854140475190*a^3 - 19881883107465415079/49270581708280950380*a^2 - 429344957526095494/12317645427070237595*a + 5279795271823257642/12317645427070237595, 1/288455737035887715314436770652170257098514870217994059753795696802268410869242240*a^31 - 20889925525748514290042805533672996475748342894658607470761/72113934258971928828609192663042564274628717554498514938448924200567102717310560*a^30 + 3088672774147775049511954201368211068485113729953924884460604901368307223161/13111624410722168877928944120553193504477948646272457261536168036466745948601920*a^29 - 17756273659347998463249243334761481153531952727910409177013513589579148912851/72113934258971928828609192663042564274628717554498514938448924200567102717310560*a^28 + 131845665850486877869498045571811334317793481460109352165014016496678927151783/144227868517943857657218385326085128549257435108997029876897848401134205434621120*a^27 - 49724992952038930985432971698292585659140495332266680755322355765239526674167/72113934258971928828609192663042564274628717554498514938448924200567102717310560*a^26 - 144126195050597204058445332843057108627859263238036317760230965847501410161209/13111624410722168877928944120553193504477948646272457261536168036466745948601920*a^25 + 5960314693012913034673771651892887841414898855475336984639833216182842830517309/288455737035887715314436770652170257098514870217994059753795696802268410869242240*a^24 - 207756391561954845273550372424884045352066416721536405919013638544379298804051/28845573703588771531443677065217025709851487021799405975379569680226841086924224*a^23 + 60984093617327819022202825678590855000205147842912088416762208028501708314727/144227868517943857657218385326085128549257435108997029876897848401134205434621120*a^22 + 1024186446197686093572325241212738706385005765168811980776040808513382748320543/57691147407177543062887354130434051419702974043598811950759139360453682173848448*a^21 + 9831042507618616027385144016903158327843023448559645241574140768170282938769579/288455737035887715314436770652170257098514870217994059753795696802268410869242240*a^20 + 36941243170342065365282645942015057089059538944322032325915769937612449078621413/288455737035887715314436770652170257098514870217994059753795696802268410869242240*a^19 - 24395308491564165579193667494171621620518431077619764138314649152808164940615/1407101156272623001533837905620342717553731074234117364652661935620821516435328*a^18 - 1267264392801332914042714759181298858546115496662351935390614126876815032467231/7035505781363115007669189528101713587768655371170586823263309678104107582176640*a^17 - 17251460453225419532445921090521100684514074184649428115996885548218958739901299/144227868517943857657218385326085128549257435108997029876897848401134205434621120*a^16 + 10725817611827609261664585017885278079043844423286202792399306676999346585646289/288455737035887715314436770652170257098514870217994059753795696802268410869242240*a^15 + 22864808203041327921020023035694648115748819663818046594345878219221148571246753/57691147407177543062887354130434051419702974043598811950759139360453682173848448*a^14 - 60625644037309444498960802201814666103985504464686599571527749466015652084493001/288455737035887715314436770652170257098514870217994059753795696802268410869242240*a^13 - 36646045520553523588319418058038254443420380187596401135895827646678438346617441/288455737035887715314436770652170257098514870217994059753795696802268410869242240*a^12 + 10281603340531812198380572742680961135125059348478133975241500885002534974358849/288455737035887715314436770652170257098514870217994059753795696802268410869242240*a^11 - 83029105081459595800697279191414515222659507651418277405637446053070992174566561/288455737035887715314436770652170257098514870217994059753795696802268410869242240*a^10 - 11699787871173739989526990738939861953508225249184242724002166841686699741289403/36056967129485964414304596331521282137314358777249257469224462100283551358655280*a^9 - 125745480692280213823497352011627512242641632106295126112583206277039550953223297/288455737035887715314436770652170257098514870217994059753795696802268410869242240*a^8 - 133551343301505639906055217392227209293363396158164686846228377816980331760800453/288455737035887715314436770652170257098514870217994059753795696802268410869242240*a^7 + 3399062661061449389750201068041464829979883463543931269165695810597979114163019/144227868517943857657218385326085128549257435108997029876897848401134205434621120*a^6 - 3034959994759567152566312187662941858361597202084590017553308950094371276544307/9014241782371491103576149082880320534328589694312314367306115525070887839663820*a^5 + 1068440590703473895857464023748608889677815118482354737478075804007806673189049/7211393425897192882860919266304256427462871755449851493844892420056710271731056*a^4 - 4131298301815072323612228344225588362907184802166146299461174998687034468904889/9014241782371491103576149082880320534328589694312314367306115525070887839663820*a^3 - 461678264904253273056718599379533103954605283210223099397210015209516688448469/9014241782371491103576149082880320534328589694312314367306115525070887839663820*a^2 + 377281144831000144994964273914202687611090811711442589308049308712410194860453/2253560445592872775894037270720080133582147423578078591826528881267721959915955*a - 902499289029543860064023555750542287594322284588310982843126081405894084785838/2253560445592872775894037270720080133582147423578078591826528881267721959915955

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

not computed

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{720893405312274704384024174360685296659707118301748692365836756778748803403}{7035505781363115007669189528101713587768655371170586823263309678104107582176640} a^{31} - \frac{596956557805948854765838033802918985478218753799444563787170535496758268653}{3517752890681557503834594764050856793884327685585293411631654839052053791088320} a^{30} - \frac{479777494337456926140951821115732645969997403064569164714690200752224402581}{319795717334687045803144978550077890353120698689572128330150439913823071917120} a^{29} + \frac{305150277902964982080450321436218825486080763764749327742502795180861052357}{109929777833798671994831086376589274808885240174540419113489213720376680971510} a^{28} + \frac{43385588945075487318662641549257719771732045915112142673300468661568652973001}{3517752890681557503834594764050856793884327685585293411631654839052053791088320} a^{27} - \frac{19524629652676093189439017191830204283900404707391320893013975636023288967357}{879438222670389375958648691012714198471081921396323352907913709763013447772080} a^{26} - \frac{65246452705993631676282428431600423903200728090584543857598436082288534426313}{703550578136311500766918952810171358776865537117058682326330967810410758217664} a^{25} + \frac{1057139610785117946265267629172011015936283835299492623258945253428503014210483}{7035505781363115007669189528101713587768655371170586823263309678104107582176640} a^{24} + \frac{50864052877728424274788011946392042436278705133253989834288543491402135351539}{87943822267038937595864869101271419847108192139632335290791370976301344777208} a^{23} - \frac{1588430148432129872380769971762083610120436105276001596479569457892433194012949}{3517752890681557503834594764050856793884327685585293411631654839052053791088320} a^{22} - \frac{25615415967767905633175080349919486927733946751837126482121395904752933010247107}{7035505781363115007669189528101713587768655371170586823263309678104107582176640} a^{21} - \frac{314522513511228076262093155603260373462475004380845358419675455367518728886851}{639591434669374091606289957100155780706241397379144256660300879827646143834240} a^{20} + \frac{105788061077183621515698657400877550192121863710169569808035404523677414585759173}{7035505781363115007669189528101713587768655371170586823263309678104107582176640} a^{19} + \frac{94497221338830408499709142479297854904052311270567200971282970091036122034311649}{7035505781363115007669189528101713587768655371170586823263309678104107582176640} a^{18} - \frac{119638420733514272990780685304610939167120321777014247366540493557393139051714163}{7035505781363115007669189528101713587768655371170586823263309678104107582176640} a^{17} - \frac{12454516549102420493126370701976242816154395605859983065069364169058296729982577}{109929777833798671994831086376589274808885240174540419113489213720376680971510} a^{16} + \frac{391591350375108354020000744113171819761768878137808464379681261148673707522516559}{7035505781363115007669189528101713587768655371170586823263309678104107582176640} a^{15} + \frac{19858911698605301473625583309729039502770814761642971884844387402279711111161965}{1407101156272623001533837905620342717553731074234117364652661935620821516435328} a^{14} + \frac{3464285953602823352464692217845569091951062010793820128039052909172814724256622223}{7035505781363115007669189528101713587768655371170586823263309678104107582176640} a^{13} - \frac{3478847387988679038819046464148892171584385035618038410001312558998836606079581837}{7035505781363115007669189528101713587768655371170586823263309678104107582176640} a^{12} + \frac{1514986832121584754699326432454806256899966699572218127257995052463790606989706697}{7035505781363115007669189528101713587768655371170586823263309678104107582176640} a^{11} - \frac{1618229660153217740034487933706175522895336443355222345676590555665802954492587573}{1407101156272623001533837905620342717553731074234117364652661935620821516435328} a^{10} + \frac{4160484814643741413561690604518680499742073474136097236885459683823111637378269411}{3517752890681557503834594764050856793884327685585293411631654839052053791088320} a^{9} - \frac{3770202377147965049975509288459902997888459012679425031916245422253191925650469003}{7035505781363115007669189528101713587768655371170586823263309678104107582176640} a^{8} + \frac{9188616291050342071131025261612071937268100375632243546483498479230230698864414847}{7035505781363115007669189528101713587768655371170586823263309678104107582176640} a^{7} - \frac{246576089343840664193652258959629536810002565021862986515535103086230369833052803}{219859555667597343989662172753178549617770480349080838226978427440753361943020} a^{6} + \frac{580191987481316088170877496167722494917407277619169600183393129350419750496597883}{1758876445340778751917297382025428396942163842792646705815827419526026895544160} a^{5} - \frac{243457380637994224799065036560752278636221878898445284265584889120018070569974627}{439719111335194687979324345506357099235540960698161676453956854881506723886040} a^{4} + \frac{147988898343866193410477705552382166395709041241326553875929475261891313868573703}{439719111335194687979324345506357099235540960698161676453956854881506723886040} a^{3} + \frac{696277462201536578645228476898001704747474893039124675576575551495364075758351}{9993616166708970181348280579689934073535021834049129010317201247306970997410} a^{2} + \frac{581495348273575027564300947129498206945572193231756198136694240971551868263301}{109929777833798671994831086376589274808885240174540419113489213720376680971510} a - \frac{18798842408387479295842271097217523103224547946640320697506112845384717720273}{54964888916899335997415543188294637404442620087270209556744606860188340485755} \) (720893405312274704384024174360685296659707118301748692365836756778748803403)/(7035505781363115007669189528101713587768655371170586823263309678104107582176640)a^(31) - (596956557805948854765838033802918985478218753799444563787170535496758268653)/(3517752890681557503834594764050856793884327685585293411631654839052053791088320)a^(30) - (479777494337456926140951821115732645969997403064569164714690200752224402581)/(319795717334687045803144978550077890353120698689572128330150439913823071917120)a^(29) + (305150277902964982080450321436218825486080763764749327742502795180861052357)/(109929777833798671994831086376589274808885240174540419113489213720376680971510)a^(28) + (43385588945075487318662641549257719771732045915112142673300468661568652973001)/(3517752890681557503834594764050856793884327685585293411631654839052053791088320)a^(27) - (19524629652676093189439017191830204283900404707391320893013975636023288967357)/(879438222670389375958648691012714198471081921396323352907913709763013447772080)a^(26) - (65246452705993631676282428431600423903200728090584543857598436082288534426313)/(703550578136311500766918952810171358776865537117058682326330967810410758217664)a^(25) + (1057139610785117946265267629172011015936283835299492623258945253428503014210483)/(7035505781363115007669189528101713587768655371170586823263309678104107582176640)a^(24) + (50864052877728424274788011946392042436278705133253989834288543491402135351539)/(87943822267038937595864869101271419847108192139632335290791370976301344777208)a^(23) - (1588430148432129872380769971762083610120436105276001596479569457892433194012949)/(3517752890681557503834594764050856793884327685585293411631654839052053791088320)a^(22) - (25615415967767905633175080349919486927733946751837126482121395904752933010247107)/(7035505781363115007669189528101713587768655371170586823263309678104107582176640)a^(21) - (314522513511228076262093155603260373462475004380845358419675455367518728886851)/(639591434669374091606289957100155780706241397379144256660300879827646143834240)a^(20) + (105788061077183621515698657400877550192121863710169569808035404523677414585759173)/(7035505781363115007669189528101713587768655371170586823263309678104107582176640)a^(19) + (94497221338830408499709142479297854904052311270567200971282970091036122034311649)/(7035505781363115007669189528101713587768655371170586823263309678104107582176640)a^(18) - (119638420733514272990780685304610939167120321777014247366540493557393139051714163)/(7035505781363115007669189528101713587768655371170586823263309678104107582176640)a^(17) - (12454516549102420493126370701976242816154395605859983065069364169058296729982577)/(109929777833798671994831086376589274808885240174540419113489213720376680971510)a^(16) + (391591350375108354020000744113171819761768878137808464379681261148673707522516559)/(7035505781363115007669189528101713587768655371170586823263309678104107582176640)a^(15) + (19858911698605301473625583309729039502770814761642971884844387402279711111161965)/(1407101156272623001533837905620342717553731074234117364652661935620821516435328)a^(14) + (3464285953602823352464692217845569091951062010793820128039052909172814724256622223)/(7035505781363115007669189528101713587768655371170586823263309678104107582176640)a^(13) - (3478847387988679038819046464148892171584385035618038410001312558998836606079581837)/(7035505781363115007669189528101713587768655371170586823263309678104107582176640)a^(12) + (1514986832121584754699326432454806256899966699572218127257995052463790606989706697)/(7035505781363115007669189528101713587768655371170586823263309678104107582176640)a^(11) - (1618229660153217740034487933706175522895336443355222345676590555665802954492587573)/(1407101156272623001533837905620342717553731074234117364652661935620821516435328)a^(10) + (4160484814643741413561690604518680499742073474136097236885459683823111637378269411)/(3517752890681557503834594764050856793884327685585293411631654839052053791088320)a^(9) - (3770202377147965049975509288459902997888459012679425031916245422253191925650469003)/(7035505781363115007669189528101713587768655371170586823263309678104107582176640)a^(8) + (9188616291050342071131025261612071937268100375632243546483498479230230698864414847)/(7035505781363115007669189528101713587768655371170586823263309678104107582176640)a^(7) - (246576089343840664193652258959629536810002565021862986515535103086230369833052803)/(219859555667597343989662172753178549617770480349080838226978427440753361943020)a^(6) + (580191987481316088170877496167722494917407277619169600183393129350419750496597883)/(1758876445340778751917297382025428396942163842792646705815827419526026895544160)a^(5) - (243457380637994224799065036560752278636221878898445284265584889120018070569974627)/(439719111335194687979324345506357099235540960698161676453956854881506723886040)a^(4) + (147988898343866193410477705552382166395709041241326553875929475261891313868573703)/(439719111335194687979324345506357099235540960698161676453956854881506723886040)a^(3) + (696277462201536578645228476898001704747474893039124675576575551495364075758351)/(9993616166708970181348280579689934073535021834049129010317201247306970997410)a^(2) + (581495348273575027564300947129498206945572193231756198136694240971551868263301)/(109929777833798671994831086376589274808885240174540419113489213720376680971510)*a - (18798842408387479295842271097217523103224547946640320697506112845384717720273)/(54964888916899335997415543188294637404442620087270209556744606860188340485755) (order $30$)	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:	not computed	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:	not computed	sage: K.regulator() gp: K.reg magma: Regulator(K); oscar: regulator(K)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr $ not computed \end{aligned}\]

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

x = polygen(QQ); K.<a> = NumberField(x^32 - 2*x^31 - 14*x^30 + 32*x^29 + 110*x^28 - 256*x^27 - 822*x^26 + 1761*x^25 + 5076*x^24 - 6234*x^23 - 33617*x^22 + 7049*x^21 + 145847*x^20 + 80587*x^19 - 200297*x^18 - 1040496*x^17 + 909785*x^16 - 126545*x^15 + 4807173*x^14 - 6468687*x^13 + 4092091*x^12 - 12326083*x^11 + 15573178*x^10 - 9966741*x^9 + 15430857*x^8 - 15762104*x^7 + 7858456*x^6 - 7358464*x^5 + 5444944*x^4 - 823424*x^3 + 88384*x^2 - 5760*x + 256)

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^31 - 14*x^30 + 32*x^29 + 110*x^28 - 256*x^27 - 822*x^26 + 1761*x^25 + 5076*x^24 - 6234*x^23 - 33617*x^22 + 7049*x^21 + 145847*x^20 + 80587*x^19 - 200297*x^18 - 1040496*x^17 + 909785*x^16 - 126545*x^15 + 4807173*x^14 - 6468687*x^13 + 4092091*x^12 - 12326083*x^11 + 15573178*x^10 - 9966741*x^9 + 15430857*x^8 - 15762104*x^7 + 7858456*x^6 - 7358464*x^5 + 5444944*x^4 - 823424*x^3 + 88384*x^2 - 5760*x + 256, 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^31 - 14*x^30 + 32*x^29 + 110*x^28 - 256*x^27 - 822*x^26 + 1761*x^25 + 5076*x^24 - 6234*x^23 - 33617*x^22 + 7049*x^21 + 145847*x^20 + 80587*x^19 - 200297*x^18 - 1040496*x^17 + 909785*x^16 - 126545*x^15 + 4807173*x^14 - 6468687*x^13 + 4092091*x^12 - 12326083*x^11 + 15573178*x^10 - 9966741*x^9 + 15430857*x^8 - 15762104*x^7 + 7858456*x^6 - 7358464*x^5 + 5444944*x^4 - 823424*x^3 + 88384*x^2 - 5760*x + 256);

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^31 - 14*x^30 + 32*x^29 + 110*x^28 - 256*x^27 - 822*x^26 + 1761*x^25 + 5076*x^24 - 6234*x^23 - 33617*x^22 + 7049*x^21 + 145847*x^20 + 80587*x^19 - 200297*x^18 - 1040496*x^17 + 909785*x^16 - 126545*x^15 + 4807173*x^14 - 6468687*x^13 + 4092091*x^12 - 12326083*x^11 + 15573178*x^10 - 9966741*x^9 + 15430857*x^8 - 15762104*x^7 + 7858456*x^6 - 7358464*x^5 + 5444944*x^4 - 823424*x^3 + 88384*x^2 - 5760*x + 256);

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

$C_2^4:C_4$ (as 32T262):

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 $C_2^4:C_4$

Character table for $C_2^4:C_4$

Intermediate fields

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	${\href{/padicField/2.4.0.1}{4} }^{8}$	R	R	R	${\href{/padicField/11.2.0.1}{2} }^{16}$	${\href{/padicField/13.4.0.1}{4} }^{8}$	${\href{/padicField/17.4.0.1}{4} }^{8}$	${\href{/padicField/19.2.0.1}{2} }^{16}$	${\href{/padicField/23.4.0.1}{4} }^{8}$	R	${\href{/padicField/31.2.0.1}{2} }^{16}$	${\href{/padicField/37.4.0.1}{4} }^{8}$	${\href{/padicField/41.2.0.1}{2} }^{16}$	${\href{/padicField/43.4.0.1}{4} }^{8}$	${\href{/padicField/47.4.0.1}{4} }^{8}$	${\href{/padicField/53.4.0.1}{4} }^{8}$	${\href{/padicField/59.2.0.1}{2} }^{16}$

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
$3$ 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	5.8.6.1	$x^{8} + 16 x^{7} + 104 x^{6} + 352 x^{5} + 674 x^{4} + 784 x^{3} + 776 x^{2} + 928 x + 721$	$4$	$2$	$6$	$C_4\times C_2$	$[\ ]_{4}^{2}$
	5.8.6.1	$x^{8} + 16 x^{7} + 104 x^{6} + 352 x^{5} + 674 x^{4} + 784 x^{3} + 776 x^{2} + 928 x + 721$	$4$	$2$	$6$	$C_4\times C_2$	$[\ ]_{4}^{2}$
	5.8.6.1	$x^{8} + 16 x^{7} + 104 x^{6} + 352 x^{5} + 674 x^{4} + 784 x^{3} + 776 x^{2} + 928 x + 721$	$4$	$2$	$6$	$C_4\times C_2$	$[\ ]_{4}^{2}$
	5.8.6.1	$x^{8} + 16 x^{7} + 104 x^{6} + 352 x^{5} + 674 x^{4} + 784 x^{3} + 776 x^{2} + 928 x + 721$	$4$	$2$	$6$	$C_4\times C_2$	$[\ ]_{4}^{2}$
$7$ 7	7.8.4.1	$x^{8} + 38 x^{6} + 8 x^{5} + 395 x^{4} - 72 x^{3} + 1026 x^{2} - 872 x + 401$	$2$	$4$	$4$	$C_4\times C_2$	$[\ ]_{2}^{4}$
	7.8.4.1	$x^{8} + 38 x^{6} + 8 x^{5} + 395 x^{4} - 72 x^{3} + 1026 x^{2} - 872 x + 401$	$2$	$4$	$4$	$C_4\times C_2$	$[\ ]_{2}^{4}$
	7.8.4.1	$x^{8} + 38 x^{6} + 8 x^{5} + 395 x^{4} - 72 x^{3} + 1026 x^{2} - 872 x + 401$	$2$	$4$	$4$	$C_4\times C_2$	$[\ ]_{2}^{4}$
	7.8.4.1	$x^{8} + 38 x^{6} + 8 x^{5} + 395 x^{4} - 72 x^{3} + 1026 x^{2} - 872 x + 401$	$2$	$4$	$4$	$C_4\times C_2$	$[\ ]_{2}^{4}$
$29$ 29	29.2.0.1	$x^{2} + 24 x + 2$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
	29.2.0.1	$x^{2} + 24 x + 2$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
	29.2.0.1	$x^{2} + 24 x + 2$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
	29.2.0.1	$x^{2} + 24 x + 2$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
	29.2.0.1	$x^{2} + 24 x + 2$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
	29.2.0.1	$x^{2} + 24 x + 2$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
	29.2.0.1	$x^{2} + 24 x + 2$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
	29.2.0.1	$x^{2} + 24 x + 2$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
	29.4.2.1	$x^{4} + 1440 x^{3} + 535166 x^{2} + 12071520 x + 1504089$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$
	29.4.2.1	$x^{4} + 1440 x^{3} + 535166 x^{2} + 12071520 x + 1504089$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$
	29.4.2.1	$x^{4} + 1440 x^{3} + 535166 x^{2} + 12071520 x + 1504089$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$
	29.4.2.1	$x^{4} + 1440 x^{3} + 535166 x^{2} + 12071520 x + 1504089$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$

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