LMFDB - Number field 40.0.2843787924946259292606529888567891936342075788742277722133996384693518336.2

Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^40 - 20*x^39 + 234*x^38 - 1976*x^37 + 13279*x^36 - 74520*x^35 + 360238*x^34 - 1530680*x^33 + 5800422*x^32 - 19809432*x^31 + 61453956*x^30 - 174207264*x^29 + 453277478*x^28 - 1086128128*x^27 + 2402433828*x^26 - 4913327328*x^25 + 9300148725*x^24 - 16301626740*x^23 + 26468749370*x^22 - 39824889160*x^21 + 55573165179*x^20 - 72059216200*x^19 + 87116562190*x^18 - 98665586280*x^17 + 105104472084*x^16 - 105116990208*x^15 + 97114298392*x^14 - 79416402656*x^13 + 55222821968*x^12 - 38173429376*x^11 + 36542373728*x^10 - 33447633536*x^9 + 14972299072*x^8 + 1667546112*x^7 - 2576594560*x^6 - 1340352000*x^5 + 1398502656*x^4 - 272152576*x^3 - 20025856*x^2 + 5699584*x + 541696)

gp: K = bnfinit(y^40 - 20*y^39 + 234*y^38 - 1976*y^37 + 13279*y^36 - 74520*y^35 + 360238*y^34 - 1530680*y^33 + 5800422*y^32 - 19809432*y^31 + 61453956*y^30 - 174207264*y^29 + 453277478*y^28 - 1086128128*y^27 + 2402433828*y^26 - 4913327328*y^25 + 9300148725*y^24 - 16301626740*y^23 + 26468749370*y^22 - 39824889160*y^21 + 55573165179*y^20 - 72059216200*y^19 + 87116562190*y^18 - 98665586280*y^17 + 105104472084*y^16 - 105116990208*y^15 + 97114298392*y^14 - 79416402656*y^13 + 55222821968*y^12 - 38173429376*y^11 + 36542373728*y^10 - 33447633536*y^9 + 14972299072*y^8 + 1667546112*y^7 - 2576594560*y^6 - 1340352000*y^5 + 1398502656*y^4 - 272152576*y^3 - 20025856*y^2 + 5699584*y + 541696, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^40 - 20*x^39 + 234*x^38 - 1976*x^37 + 13279*x^36 - 74520*x^35 + 360238*x^34 - 1530680*x^33 + 5800422*x^32 - 19809432*x^31 + 61453956*x^30 - 174207264*x^29 + 453277478*x^28 - 1086128128*x^27 + 2402433828*x^26 - 4913327328*x^25 + 9300148725*x^24 - 16301626740*x^23 + 26468749370*x^22 - 39824889160*x^21 + 55573165179*x^20 - 72059216200*x^19 + 87116562190*x^18 - 98665586280*x^17 + 105104472084*x^16 - 105116990208*x^15 + 97114298392*x^14 - 79416402656*x^13 + 55222821968*x^12 - 38173429376*x^11 + 36542373728*x^10 - 33447633536*x^9 + 14972299072*x^8 + 1667546112*x^7 - 2576594560*x^6 - 1340352000*x^5 + 1398502656*x^4 - 272152576*x^3 - 20025856*x^2 + 5699584*x + 541696);

oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^40 - 20*x^39 + 234*x^38 - 1976*x^37 + 13279*x^36 - 74520*x^35 + 360238*x^34 - 1530680*x^33 + 5800422*x^32 - 19809432*x^31 + 61453956*x^30 - 174207264*x^29 + 453277478*x^28 - 1086128128*x^27 + 2402433828*x^26 - 4913327328*x^25 + 9300148725*x^24 - 16301626740*x^23 + 26468749370*x^22 - 39824889160*x^21 + 55573165179*x^20 - 72059216200*x^19 + 87116562190*x^18 - 98665586280*x^17 + 105104472084*x^16 - 105116990208*x^15 + 97114298392*x^14 - 79416402656*x^13 + 55222821968*x^12 - 38173429376*x^11 + 36542373728*x^10 - 33447633536*x^9 + 14972299072*x^8 + 1667546112*x^7 - 2576594560*x^6 - 1340352000*x^5 + 1398502656*x^4 - 272152576*x^3 - 20025856*x^2 + 5699584*x + 541696)

$ x^{40} - 20 x^{39} + 234 x^{38} - 1976 x^{37} + 13279 x^{36} - 74520 x^{35} + 360238 x^{34} + \cdots + 541696 $ x^40 - 20*x^39 + 234*x^38 - 1976*x^37 + 13279*x^36 - 74520*x^35 + 360238*x^34 - 1530680*x^33 + 5800422*x^32 - 19809432*x^31 + 61453956*x^30 - 174207264*x^29 + 453277478*x^28 - 1086128128*x^27 + 2402433828*x^26 - 4913327328*x^25 + 9300148725*x^24 - 16301626740*x^23 + 26468749370*x^22 - 39824889160*x^21 + 55573165179*x^20 - 72059216200*x^19 + 87116562190*x^18 - 98665586280*x^17 + 105104472084*x^16 - 105116990208*x^15 + 97114298392*x^14 - 79416402656*x^13 + 55222821968*x^12 - 38173429376*x^11 + 36542373728*x^10 - 33447633536*x^9 + 14972299072*x^8 + 1667546112*x^7 - 2576594560*x^6 - 1340352000*x^5 + 1398502656*x^4 - 272152576*x^3 - 20025856*x^2 + 5699584*x + 541696

sage: K.defining_polynomial()

gp: K.pol

magma: DefiningPolynomial(K);

oscar: defining_polynomial(K)

Invariants

Degree:	$40$	sage: K.degree() gp: poldegree(K.pol) magma: Degree(K); oscar: degree(K)
Signature:	$[0, 20]$	sage: K.signature() gp: K.sign magma: Signature(K); oscar: signature(K)
Discriminant:	$2843787924946259292606529888567891936342075788742277722133996384693518336$ 2843787924946259292606529888567891936342075788742277722133996384693518336 $\medspace = 2^{60}\cdot 7^{20}\cdot 11^{36}$	sage: K.disc() gp: K.disc magma: OK := Integers(K); Discriminant(OK); oscar: OK = ring_of_integers(K); discriminant(OK)
Root discriminant:	$64.77$	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:	$2^{3/2}7^{1/2}11^{9/10}\approx 64.76605288693868$
Ramified primes:	$2$, $7$, $11$ 2, 7, 11	sage: K.disc().support() gp: factor(abs(K.disc))[,1]~ magma: PrimeDivisors(Discriminant(OK)); oscar: prime_divisors(discriminant((OK)))
Discriminant root field:	$\Q$
$\card{ \Gal(K/\Q) }$:	$40$	sage: K.automorphisms() magma: Automorphisms(K); oscar: automorphisms(K)
This field is Galois and abelian over $\Q$.
Conductor:	$616=2^{3}\cdot 7\cdot 11$
Dirichlet character group:	$\lbrace$$\chi_{616}(1,·)$, $\chi_{616}(321,·)$, $\chi_{616}(393,·)$, $\chi_{616}(13,·)$, $\chi_{616}(237,·)$, $\chi_{616}(141,·)$, $\chi_{616}(405,·)$, $\chi_{616}(281,·)$, $\chi_{616}(153,·)$, $\chi_{616}(29,·)$, $\chi_{616}(197,·)$, $\chi_{616}(545,·)$, $\chi_{616}(293,·)$, $\chi_{616}(41,·)$, $\chi_{616}(349,·)$, $\chi_{616}(433,·)$, $\chi_{616}(309,·)$, $\chi_{616}(265,·)$, $\chi_{616}(57,·)$, $\chi_{616}(573,·)$, $\chi_{616}(181,·)$, $\chi_{616}(449,·)$, $\chi_{616}(69,·)$, $\chi_{616}(97,·)$, $\chi_{616}(461,·)$, $\chi_{616}(589,·)$, $\chi_{616}(337,·)$, $\chi_{616}(85,·)$, $\chi_{616}(377,·)$, $\chi_{616}(601,·)$, $\chi_{616}(477,·)$, $\chi_{616}(421,·)$, $\chi_{616}(225,·)$, $\chi_{616}(489,·)$, $\chi_{616}(365,·)$, $\chi_{616}(113,·)$, $\chi_{616}(169,·)$, $\chi_{616}(505,·)$, $\chi_{616}(125,·)$, $\chi_{616}(533,·)$$\rbrace$
This is a CM field.
Reflex fields:	unavailable$^{524288}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2}a^{4}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{5}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{3}$, $\frac{1}{4}a^{8}-\frac{1}{4}a^{4}$, $\frac{1}{4}a^{9}-\frac{1}{4}a^{5}$, $\frac{1}{4}a^{10}-\frac{1}{4}a^{6}$, $\frac{1}{4}a^{11}-\frac{1}{4}a^{7}$, $\frac{1}{8}a^{12}-\frac{1}{8}a^{10}-\frac{1}{8}a^{8}+\frac{1}{8}a^{6}$, $\frac{1}{8}a^{13}-\frac{1}{8}a^{11}-\frac{1}{8}a^{9}+\frac{1}{8}a^{7}$, $\frac{1}{8}a^{14}-\frac{1}{8}a^{6}$, $\frac{1}{8}a^{15}-\frac{1}{8}a^{7}$, $\frac{1}{16}a^{16}-\frac{1}{8}a^{10}-\frac{1}{16}a^{8}+\frac{1}{8}a^{6}$, $\frac{1}{16}a^{17}-\frac{1}{8}a^{11}-\frac{1}{16}a^{9}+\frac{1}{8}a^{7}$, $\frac{1}{16}a^{18}+\frac{1}{16}a^{10}-\frac{1}{8}a^{6}$, $\frac{1}{16}a^{19}+\frac{1}{16}a^{11}-\frac{1}{8}a^{7}$, $\frac{1}{32}a^{20}-\frac{1}{32}a^{18}-\frac{1}{16}a^{14}+\frac{1}{32}a^{12}+\frac{3}{32}a^{10}-\frac{1}{16}a^{8}$, $\frac{1}{32}a^{21}-\frac{1}{32}a^{19}-\frac{1}{16}a^{15}+\frac{1}{32}a^{13}+\frac{3}{32}a^{11}-\frac{1}{16}a^{9}$, $\frac{1}{32}a^{22}-\frac{1}{32}a^{18}-\frac{1}{32}a^{14}+\frac{1}{32}a^{10}$, $\frac{1}{736}a^{23}-\frac{1}{32}a^{19}+\frac{1}{32}a^{15}-\frac{3}{32}a^{11}-\frac{1}{8}a^{7}+\frac{5}{23}a$, $\frac{1}{1472}a^{24}-\frac{1}{64}a^{20}+\frac{1}{64}a^{16}-\frac{3}{64}a^{12}-\frac{1}{8}a^{10}-\frac{1}{16}a^{8}+\frac{1}{8}a^{6}+\frac{5}{46}a^{2}$, $\frac{1}{1472}a^{25}-\frac{1}{64}a^{21}+\frac{1}{64}a^{17}-\frac{3}{64}a^{13}-\frac{1}{8}a^{11}-\frac{1}{16}a^{9}+\frac{1}{8}a^{7}+\frac{5}{46}a^{3}$, $\frac{1}{1472}a^{26}-\frac{1}{64}a^{22}+\frac{1}{64}a^{18}-\frac{3}{64}a^{14}+\frac{1}{16}a^{10}-\frac{1}{8}a^{6}+\frac{5}{46}a^{4}$, $\frac{1}{1472}a^{27}-\frac{1}{1472}a^{23}-\frac{1}{64}a^{19}+\frac{3}{64}a^{15}+\frac{3}{32}a^{11}-\frac{1}{8}a^{7}+\frac{5}{46}a^{5}-\frac{1}{2}a^{3}+\frac{9}{23}a$, $\frac{1}{2944}a^{28}-\frac{1}{2944}a^{26}-\frac{1}{2944}a^{24}+\frac{1}{128}a^{22}-\frac{1}{128}a^{20}+\frac{3}{128}a^{18}+\frac{3}{128}a^{16}+\frac{3}{128}a^{14}+\frac{3}{64}a^{12}-\frac{1}{16}a^{8}+\frac{5}{92}a^{6}+\frac{9}{46}a^{4}-\frac{7}{23}a^{2}$, $\frac{1}{2944}a^{29}-\frac{1}{2944}a^{27}-\frac{1}{2944}a^{25}-\frac{1}{2944}a^{23}-\frac{1}{128}a^{21}+\frac{3}{128}a^{19}+\frac{3}{128}a^{17}-\frac{5}{128}a^{15}+\frac{3}{64}a^{13}-\frac{1}{8}a^{11}-\frac{1}{16}a^{9}+\frac{5}{92}a^{7}+\frac{9}{46}a^{5}+\frac{9}{46}a^{3}-\frac{7}{23}a$, $\frac{1}{2944}a^{30}-\frac{1}{64}a^{22}+\frac{3}{128}a^{14}-\frac{1}{16}a^{10}+\frac{5}{92}a^{8}-\frac{1}{4}a^{6}-\frac{1}{4}a^{4}-\frac{1}{2}a^{2}$, $\frac{1}{2944}a^{31}-\frac{1}{1472}a^{23}-\frac{1}{32}a^{19}-\frac{1}{128}a^{15}-\frac{1}{32}a^{11}+\frac{5}{92}a^{9}-\frac{1}{8}a^{7}-\frac{1}{4}a^{5}+\frac{9}{23}a$, $\frac{1}{5888}a^{32}-\frac{1}{2944}a^{24}-\frac{1}{64}a^{20}-\frac{1}{32}a^{18}-\frac{1}{256}a^{16}-\frac{1}{16}a^{14}-\frac{1}{64}a^{12}-\frac{3}{736}a^{10}-\frac{1}{16}a^{8}-\frac{1}{4}a^{6}-\frac{1}{4}a^{4}-\frac{7}{23}a^{2}$, $\frac{1}{5888}a^{33}-\frac{1}{2944}a^{25}-\frac{1}{64}a^{21}-\frac{1}{32}a^{19}-\frac{1}{256}a^{17}-\frac{1}{16}a^{15}-\frac{1}{64}a^{13}-\frac{3}{736}a^{11}-\frac{1}{16}a^{9}-\frac{1}{4}a^{7}-\frac{1}{4}a^{5}-\frac{7}{23}a^{3}$, $\frac{1}{5888}a^{34}-\frac{1}{2944}a^{26}-\frac{1}{64}a^{22}+\frac{7}{256}a^{18}+\frac{3}{64}a^{14}+\frac{5}{184}a^{12}-\frac{1}{32}a^{10}-\frac{1}{8}a^{8}+\frac{1}{8}a^{6}-\frac{5}{92}a^{4}$, $\frac{1}{5888}a^{35}-\frac{1}{2944}a^{27}-\frac{1}{1472}a^{23}-\frac{1}{256}a^{19}+\frac{1}{64}a^{15}+\frac{5}{184}a^{13}-\frac{1}{8}a^{9}-\frac{1}{4}a^{7}-\frac{5}{92}a^{5}+\frac{9}{23}a$, $\frac{1}{11776}a^{36}-\frac{1}{11776}a^{34}-\frac{1}{5888}a^{28}+\frac{1}{5888}a^{26}-\frac{1}{2944}a^{24}-\frac{1}{128}a^{22}-\frac{1}{512}a^{20}+\frac{1}{512}a^{18}+\frac{1}{128}a^{16}-\frac{167}{2944}a^{14}-\frac{5}{368}a^{12}-\frac{1}{16}a^{10}+\frac{1}{16}a^{8}-\frac{5}{184}a^{6}-\frac{9}{92}a^{4}+\frac{9}{46}a^{2}$, $\frac{1}{11776}a^{37}-\frac{1}{11776}a^{35}-\frac{1}{5888}a^{29}+\frac{1}{5888}a^{27}-\frac{1}{2944}a^{25}+\frac{1}{2944}a^{23}-\frac{1}{512}a^{21}+\frac{1}{512}a^{19}+\frac{1}{128}a^{17}+\frac{17}{2944}a^{15}-\frac{5}{368}a^{13}+\frac{1}{16}a^{11}+\frac{1}{16}a^{9}-\frac{5}{184}a^{7}-\frac{9}{92}a^{5}-\frac{7}{23}a^{3}+\frac{7}{23}a$, $\frac{1}{49\!\cdots\!04}a^{38}-\frac{19}{49\!\cdots\!04}a^{37}+\frac{26\!\cdots\!21}{62\!\cdots\!88}a^{36}-\frac{38\!\cdots\!83}{49\!\cdots\!04}a^{35}+\frac{25\!\cdots\!19}{49\!\cdots\!04}a^{34}+\frac{15\!\cdots\!05}{62\!\cdots\!88}a^{33}-\frac{18\!\cdots\!33}{24\!\cdots\!52}a^{32}+\frac{30\!\cdots\!87}{12\!\cdots\!76}a^{31}+\frac{10\!\cdots\!67}{24\!\cdots\!52}a^{30}+\frac{55\!\cdots\!93}{24\!\cdots\!52}a^{29}-\frac{16\!\cdots\!15}{12\!\cdots\!76}a^{28}+\frac{31\!\cdots\!37}{24\!\cdots\!52}a^{27}-\frac{32\!\cdots\!03}{24\!\cdots\!52}a^{26}-\frac{20\!\cdots\!33}{62\!\cdots\!88}a^{25}+\frac{12\!\cdots\!41}{62\!\cdots\!88}a^{24}-\frac{16\!\cdots\!75}{62\!\cdots\!88}a^{23}+\frac{21\!\cdots\!51}{21\!\cdots\!48}a^{22}-\frac{45\!\cdots\!69}{21\!\cdots\!48}a^{21}-\frac{51\!\cdots\!03}{53\!\cdots\!12}a^{20}+\frac{24\!\cdots\!99}{21\!\cdots\!48}a^{19}-\frac{21\!\cdots\!27}{21\!\cdots\!48}a^{18}+\frac{40\!\cdots\!99}{26\!\cdots\!56}a^{17}-\frac{47\!\cdots\!87}{24\!\cdots\!52}a^{16}-\frac{71\!\cdots\!43}{12\!\cdots\!76}a^{15}+\frac{53\!\cdots\!57}{31\!\cdots\!44}a^{14}+\frac{14\!\cdots\!39}{62\!\cdots\!88}a^{13}+\frac{60\!\cdots\!87}{62\!\cdots\!88}a^{12}+\frac{22\!\cdots\!63}{19\!\cdots\!34}a^{11}+\frac{38\!\cdots\!17}{77\!\cdots\!36}a^{10}-\frac{87\!\cdots\!29}{15\!\cdots\!72}a^{9}-\frac{89\!\cdots\!07}{77\!\cdots\!36}a^{8}-\frac{79\!\cdots\!37}{38\!\cdots\!68}a^{7}+\frac{81\!\cdots\!05}{38\!\cdots\!68}a^{6}+\frac{53\!\cdots\!57}{38\!\cdots\!68}a^{5}+\frac{94\!\cdots\!89}{38\!\cdots\!68}a^{4}+\frac{82\!\cdots\!06}{96\!\cdots\!17}a^{3}-\frac{17\!\cdots\!85}{96\!\cdots\!17}a^{2}-\frac{32\!\cdots\!45}{96\!\cdots\!17}a+\frac{20\!\cdots\!17}{42\!\cdots\!79}$, $\frac{1}{15\!\cdots\!72}a^{39}+\frac{3828713}{38\!\cdots\!68}a^{38}+\frac{41\!\cdots\!43}{15\!\cdots\!72}a^{37}+\frac{42\!\cdots\!01}{15\!\cdots\!72}a^{36}+\frac{15\!\cdots\!79}{76\!\cdots\!36}a^{35}-\frac{35\!\cdots\!45}{15\!\cdots\!72}a^{34}-\frac{13\!\cdots\!31}{11\!\cdots\!24}a^{33}-\frac{11\!\cdots\!71}{19\!\cdots\!84}a^{32}-\frac{56\!\cdots\!91}{76\!\cdots\!36}a^{31}+\frac{45\!\cdots\!15}{38\!\cdots\!68}a^{30}-\frac{49\!\cdots\!99}{76\!\cdots\!36}a^{29}-\frac{75\!\cdots\!89}{76\!\cdots\!36}a^{28}-\frac{34\!\cdots\!29}{19\!\cdots\!84}a^{27}-\frac{21\!\cdots\!39}{76\!\cdots\!36}a^{26}-\frac{10\!\cdots\!75}{38\!\cdots\!68}a^{25}-\frac{11\!\cdots\!99}{16\!\cdots\!16}a^{24}+\frac{36\!\cdots\!89}{15\!\cdots\!72}a^{23}+\frac{63\!\cdots\!29}{10\!\cdots\!76}a^{22}+\frac{93\!\cdots\!17}{66\!\cdots\!64}a^{21}+\frac{94\!\cdots\!75}{66\!\cdots\!64}a^{20}-\frac{98\!\cdots\!49}{33\!\cdots\!32}a^{19}-\frac{13\!\cdots\!79}{66\!\cdots\!64}a^{18}-\frac{98\!\cdots\!99}{38\!\cdots\!68}a^{17}+\frac{79\!\cdots\!29}{38\!\cdots\!68}a^{16}-\frac{45\!\cdots\!75}{38\!\cdots\!68}a^{15}-\frac{26\!\cdots\!35}{19\!\cdots\!84}a^{14}+\frac{25\!\cdots\!13}{19\!\cdots\!84}a^{13}+\frac{48\!\cdots\!31}{95\!\cdots\!92}a^{12}+\frac{18\!\cdots\!23}{95\!\cdots\!92}a^{11}-\frac{15\!\cdots\!09}{23\!\cdots\!48}a^{10}-\frac{27\!\cdots\!91}{23\!\cdots\!48}a^{9}-\frac{12\!\cdots\!15}{47\!\cdots\!96}a^{8}+\frac{25\!\cdots\!43}{23\!\cdots\!48}a^{7}+\frac{20\!\cdots\!59}{23\!\cdots\!48}a^{6}-\frac{18\!\cdots\!47}{59\!\cdots\!62}a^{5}-\frac{22\!\cdots\!41}{11\!\cdots\!24}a^{4}+\frac{23\!\cdots\!31}{59\!\cdots\!62}a^{3}-\frac{34\!\cdots\!61}{25\!\cdots\!94}a^{2}+\frac{11\!\cdots\!70}{29\!\cdots\!31}a-\frac{30\!\cdots\!79}{12\!\cdots\!97}$ 1, a, a^2, a^3, 1/2*a^4 - 1/2*a^2, 1/2*a^5 - 1/2*a^3, 1/2*a^6 - 1/2*a^2, 1/2*a^7 - 1/2*a^3, 1/4*a^8 - 1/4*a^4, 1/4*a^9 - 1/4*a^5, 1/4*a^10 - 1/4*a^6, 1/4*a^11 - 1/4*a^7, 1/8*a^12 - 1/8*a^10 - 1/8*a^8 + 1/8*a^6, 1/8*a^13 - 1/8*a^11 - 1/8*a^9 + 1/8*a^7, 1/8*a^14 - 1/8*a^6, 1/8*a^15 - 1/8*a^7, 1/16*a^16 - 1/8*a^10 - 1/16*a^8 + 1/8*a^6, 1/16*a^17 - 1/8*a^11 - 1/16*a^9 + 1/8*a^7, 1/16*a^18 + 1/16*a^10 - 1/8*a^6, 1/16*a^19 + 1/16*a^11 - 1/8*a^7, 1/32*a^20 - 1/32*a^18 - 1/16*a^14 + 1/32*a^12 + 3/32*a^10 - 1/16*a^8, 1/32*a^21 - 1/32*a^19 - 1/16*a^15 + 1/32*a^13 + 3/32*a^11 - 1/16*a^9, 1/32*a^22 - 1/32*a^18 - 1/32*a^14 + 1/32*a^10, 1/736*a^23 - 1/32*a^19 + 1/32*a^15 - 3/32*a^11 - 1/8*a^7 + 5/23*a, 1/1472*a^24 - 1/64*a^20 + 1/64*a^16 - 3/64*a^12 - 1/8*a^10 - 1/16*a^8 + 1/8*a^6 + 5/46*a^2, 1/1472*a^25 - 1/64*a^21 + 1/64*a^17 - 3/64*a^13 - 1/8*a^11 - 1/16*a^9 + 1/8*a^7 + 5/46*a^3, 1/1472*a^26 - 1/64*a^22 + 1/64*a^18 - 3/64*a^14 + 1/16*a^10 - 1/8*a^6 + 5/46*a^4, 1/1472*a^27 - 1/1472*a^23 - 1/64*a^19 + 3/64*a^15 + 3/32*a^11 - 1/8*a^7 + 5/46*a^5 - 1/2*a^3 + 9/23*a, 1/2944*a^28 - 1/2944*a^26 - 1/2944*a^24 + 1/128*a^22 - 1/128*a^20 + 3/128*a^18 + 3/128*a^16 + 3/128*a^14 + 3/64*a^12 - 1/16*a^8 + 5/92*a^6 + 9/46*a^4 - 7/23*a^2, 1/2944*a^29 - 1/2944*a^27 - 1/2944*a^25 - 1/2944*a^23 - 1/128*a^21 + 3/128*a^19 + 3/128*a^17 - 5/128*a^15 + 3/64*a^13 - 1/8*a^11 - 1/16*a^9 + 5/92*a^7 + 9/46*a^5 + 9/46*a^3 - 7/23*a, 1/2944*a^30 - 1/64*a^22 + 3/128*a^14 - 1/16*a^10 + 5/92*a^8 - 1/4*a^6 - 1/4*a^4 - 1/2*a^2, 1/2944*a^31 - 1/1472*a^23 - 1/32*a^19 - 1/128*a^15 - 1/32*a^11 + 5/92*a^9 - 1/8*a^7 - 1/4*a^5 + 9/23*a, 1/5888*a^32 - 1/2944*a^24 - 1/64*a^20 - 1/32*a^18 - 1/256*a^16 - 1/16*a^14 - 1/64*a^12 - 3/736*a^10 - 1/16*a^8 - 1/4*a^6 - 1/4*a^4 - 7/23*a^2, 1/5888*a^33 - 1/2944*a^25 - 1/64*a^21 - 1/32*a^19 - 1/256*a^17 - 1/16*a^15 - 1/64*a^13 - 3/736*a^11 - 1/16*a^9 - 1/4*a^7 - 1/4*a^5 - 7/23*a^3, 1/5888*a^34 - 1/2944*a^26 - 1/64*a^22 + 7/256*a^18 + 3/64*a^14 + 5/184*a^12 - 1/32*a^10 - 1/8*a^8 + 1/8*a^6 - 5/92*a^4, 1/5888*a^35 - 1/2944*a^27 - 1/1472*a^23 - 1/256*a^19 + 1/64*a^15 + 5/184*a^13 - 1/8*a^9 - 1/4*a^7 - 5/92*a^5 + 9/23*a, 1/11776*a^36 - 1/11776*a^34 - 1/5888*a^28 + 1/5888*a^26 - 1/2944*a^24 - 1/128*a^22 - 1/512*a^20 + 1/512*a^18 + 1/128*a^16 - 167/2944*a^14 - 5/368*a^12 - 1/16*a^10 + 1/16*a^8 - 5/184*a^6 - 9/92*a^4 + 9/46*a^2, 1/11776*a^37 - 1/11776*a^35 - 1/5888*a^29 + 1/5888*a^27 - 1/2944*a^25 + 1/2944*a^23 - 1/512*a^21 + 1/512*a^19 + 1/128*a^17 + 17/2944*a^15 - 5/368*a^13 + 1/16*a^11 + 1/16*a^9 - 5/184*a^7 - 9/92*a^5 - 7/23*a^3 + 7/23*a, 1/49648390291342237038180103685652778429692593009038230306528697147904*a^38 - 19/49648390291342237038180103685652778429692593009038230306528697147904*a^37 + 261103973627080876037362088349803592540714958879378547052814321/6206048786417779629772512960706597303711574126129778788316087143488*a^36 - 3870446189159539465833874631574992849169616436620843447800438683/49648390291342237038180103685652778429692593009038230306528697147904*a^35 + 2501101544135177759465562495975332251444590221306416365433534319/49648390291342237038180103685652778429692593009038230306528697147904*a^34 + 150568776107529090648311098009063016013055205720052267392678505/6206048786417779629772512960706597303711574126129778788316087143488*a^33 - 1891229502407370578461148033582660132353790667931660645408959533/24824195145671118519090051842826389214846296504519115153264348573952*a^32 + 304897383535169710196715519512502752812410075816789488728857487/12412097572835559259545025921413194607423148252259557576632174286976*a^31 + 1001555951293818256625414205610463285469446950666207094783492967/24824195145671118519090051842826389214846296504519115153264348573952*a^30 + 55111652077122190543508385868327433467006735620574711376937693/24824195145671118519090051842826389214846296504519115153264348573952*a^29 - 1634469129107170594432027198535787386379002884192386605592323715/12412097572835559259545025921413194607423148252259557576632174286976*a^28 + 316547616637368281729386856025414242007226090402319804826428237/24824195145671118519090051842826389214846296504519115153264348573952*a^27 - 3228411412367911014502849015646881109044063248933691878547070503/24824195145671118519090051842826389214846296504519115153264348573952*a^26 - 2078860250259924722001830018569372060850830009527122053241181833/6206048786417779629772512960706597303711574126129778788316087143488*a^25 + 1299365620230007597456087367303743200481858357004510503365322141/6206048786417779629772512960706597303711574126129778788316087143488*a^24 - 1656039264909168961895413623870257342042528594182269129865996275/6206048786417779629772512960706597303711574126129778788316087143488*a^23 + 21659395380162877201048789186431787066987034727715702153120531751/2158625664840966827746961029810990366508373609088618708979508571648*a^22 - 4532882903434354358230037266808024375320722701068789895312277169/2158625664840966827746961029810990366508373609088618708979508571648*a^21 - 5195244336317072260869038188789687943653777529669350673437444603/539656416210241706936740257452747591627093402272154677244877142912*a^20 + 24371398483848954642033788169203830007734490035048464517273276399/2158625664840966827746961029810990366508373609088618708979508571648*a^19 - 21968422750451824126594154546181056343345448918826702478288439527/2158625664840966827746961029810990366508373609088618708979508571648*a^18 + 4054414622134663511253051509745443241636240643354884406351825299/269828208105120853468370128726373795813546701136077338622438571456*a^17 - 47622464422224244940823856179912021912970640307843476754002768487/24824195145671118519090051842826389214846296504519115153264348573952*a^16 - 713367982403679964198338300733554629000261459869424471885476167143/12412097572835559259545025921413194607423148252259557576632174286976*a^15 + 53327376336133354080455213658764844414590743999117047590591478857/3103024393208889814886256480353298651855787063064889394158043571744*a^14 + 148348446751251146240661642921646653363703497075540091923916240039/6206048786417779629772512960706597303711574126129778788316087143488*a^13 + 60453751175070609981504201032661805225946185323736915855340639687/6206048786417779629772512960706597303711574126129778788316087143488*a^12 + 22023248408960383842575605863581608145121540693877600139692958363/193939024575555613430391030022081165740986691441555587134877723234*a^11 + 38851636291639338624377092156450047546719989200617788492618416617/775756098302222453721564120088324662963946765766222348539510892936*a^10 - 8732915577970571733344550715062491386757867505445005228305001529/1551512196604444907443128240176649325927893531532444697079021785872*a^9 - 89035458099092630102008773729545440355922139615896092804950834707/775756098302222453721564120088324662963946765766222348539510892936*a^8 - 79480070256525564903934511176406321632030140221377021332949863637/387878049151111226860782060044162331481973382883111174269755446468*a^7 + 81588978631607476681332601694584466464485284867892075146308132705/387878049151111226860782060044162331481973382883111174269755446468*a^6 + 53982740072736720782067468788516141294989362672857630378668666757/387878049151111226860782060044162331481973382883111174269755446468*a^5 + 94481239301681033352685533497579858584141871983848323638078506789/387878049151111226860782060044162331481973382883111174269755446468*a^4 + 8252592489092216338762653141830125710216609297659728395255971406/96969512287777806715195515011040582870493345720777793567438861617*a^3 - 17722788839113486007803037754558391365149723851768469813950064485/96969512287777806715195515011040582870493345720777793567438861617*a^2 - 32877184011254709362580550981968142425111955966061750844380261845/96969512287777806715195515011040582870493345720777793567438861617*a + 204050938334667692032162228659758778444656121124516984932532217/4216065751642513335443283261349590559586667205251208415975602679, 1/1520717434987507845524537763604897390537427692870437671463785215765132508672*a^39 + 3828713/380179358746876961381134440901224347634356923217609417865946303941283127168*a^38 + 41330752358440603476077843262095513590184134458881286325350101387786943/1520717434987507845524537763604897390537427692870437671463785215765132508672*a^37 + 42193334466225964017720280535662863113783593757856519889406385225853501/1520717434987507845524537763604897390537427692870437671463785215765132508672*a^36 + 15662055336914372626652001965276265477557722765981856386783032297049879/760358717493753922762268881802448695268713846435218835731892607882566254336*a^35 - 35777058156829078815504061149740796077201438125270085884214470799714245/1520717434987507845524537763604897390537427692870437671463785215765132508672*a^34 - 130123822995005551529213125992461836370521070469066492548030493312131/11880604960839905043160451278163260863573653850550294308310821998165097724*a^33 - 11127126609071993704516948497034522222043343250327548657052946869458271/190089679373438480690567220450612173817178461608804708932973151970641563584*a^32 - 5611779616027315085830396949524970168488768513307405223442824411805591/760358717493753922762268881802448695268713846435218835731892607882566254336*a^31 + 45534511210251055955142450107502542417901486016015473645457783304502615/380179358746876961381134440901224347634356923217609417865946303941283127168*a^30 - 49467839553326965891682080332987580239140109499639764206707123831607699/760358717493753922762268881802448695268713846435218835731892607882566254336*a^29 - 75811825358978905882610618071259581241910054588765695758277011840156389/760358717493753922762268881802448695268713846435218835731892607882566254336*a^28 - 34506496215821022391175590830752173677157063796645530943568634317125729/190089679373438480690567220450612173817178461608804708932973151970641563584*a^27 - 218499301328374667998374314478711533296985270092843548570739352830717739/760358717493753922762268881802448695268713846435218835731892607882566254336*a^26 - 101819168782842828097232035455748297518068696666718891695335109511480775/380179358746876961381134440901224347634356923217609417865946303941283127168*a^25 - 1116786885784292489053792119256335007020714136881384612047263458303899/16529537336820737451353671343531493375406822748591713820258534953968831616*a^24 + 366020627459422669103221752380400122957671994922083729549132766834324689/1520717434987507845524537763604897390537427692870437671463785215765132508672*a^23 + 6359592044637756424259083508726988547139913329370973587022055499075729/1033096083551296090709604458970718335962926421786982113766158434623051976*a^22 + 934071967971594060130426023360916684463001737889833567687829427835694817/66118149347282949805414685374125973501627290994366855281034139815875326464*a^21 + 949178387887019573726791785194172388218756075632442914952938728556784675/66118149347282949805414685374125973501627290994366855281034139815875326464*a^20 - 982977268737960468138444395353869456061544116515351823155743507659856049/33059074673641474902707342687062986750813645497183427640517069907937663232*a^19 - 1314045148660860716251427716940193853549202209010128136319578980572301979/66118149347282949805414685374125973501627290994366855281034139815875326464*a^18 - 983450067219716358669932186143348206391602106068250568767940166708270399/380179358746876961381134440901224347634356923217609417865946303941283127168*a^17 + 7912869717556981625979417806124793129872770967147739957243768701431556529/380179358746876961381134440901224347634356923217609417865946303941283127168*a^16 - 4576569950450107664010582232363644084024171720806206259651698293334786075/380179358746876961381134440901224347634356923217609417865946303941283127168*a^15 - 2664824058614304793602825060087693009920402852402585327413995749938657735/190089679373438480690567220450612173817178461608804708932973151970641563584*a^14 + 254659469375013864762861779770483200157929848568249901983781298525868113/190089679373438480690567220450612173817178461608804708932973151970641563584*a^13 + 4857731572772213076343059681360486857469683747010641956154592729624243131/95044839686719240345283610225306086908589230804402354466486575985320781792*a^12 + 1878819730136845746367247826435859410022744541701621178024727930096574123/95044839686719240345283610225306086908589230804402354466486575985320781792*a^11 - 154510582888982461720363240442671765907164593617877054516576891527345009/23761209921679810086320902556326521727147307701100588616621643996330195448*a^10 - 2782969043355821131130679487306970066764309619425878796032990519300563191/23761209921679810086320902556326521727147307701100588616621643996330195448*a^9 - 1256092802329024529214692839152121097582773788764228231501939258778480415/47522419843359620172641805112653043454294615402201177233243287992660390896*a^8 + 2526087512237189556580250764538432192291947108214120514735070102606143143/23761209921679810086320902556326521727147307701100588616621643996330195448*a^7 + 2043296412639449680706291608116952909951634345858240603819608283889662959/23761209921679810086320902556326521727147307701100588616621643996330195448*a^6 - 181517416854905958282639361288086066248623538936486531494373587770884747/5940302480419952521580225639081630431786826925275147154155410999082548862*a^5 - 2286280184163032100156326481279367513955400962562390733263530359332482941/11880604960839905043160451278163260863573653850550294308310821998165097724*a^4 + 2362433464799583044219214608805823432965136345289303417740287227361733731/5940302480419952521580225639081630431786826925275147154155410999082548862*a^3 - 34967113692951015744720905801619905118984924936555977963760947192372061/258274020887824022677401114742679583990731605446745528441539608655762994*a^2 + 1173933748980493464786841409224870186110190024522153066894479816731582670/2970151240209976260790112819540815215893413462637573577077705499541274431*a - 30811507602907247905096732644215436594362497147531177882247887207866779/129137010443912011338700557371339791995365802723372764220769804327881497

sage: K.integral_basis()

gp: K.zk

magma: IntegralBasis(K);

oscar: basis(OK)

Monogenic:		No
Index:		Not computed
Inessential primes:		$23$

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:	$19$	sage: UK.rank() gp: K.fu magma: UnitRank(K); oscar: rank(UK)
Torsion generator:	\( \frac{15041490416277220308715037744496914119905014927547888601830}{96969512287777806715195515011040582870493345720777793567438861617} a^{38} - \frac{285788317909267185865585717145441368278195283623409883434770}{96969512287777806715195515011040582870493345720777793567438861617} a^{37} + \frac{3236868595667634317862414339534700504295561515362787074414765}{96969512287777806715195515011040582870493345720777793567438861617} a^{36} - \frac{26541131434088760090443443508480617198440430794331670278206300}{96969512287777806715195515011040582870493345720777793567438861617} a^{35} + \frac{173828646740518385726063441927098406984786359642476106328063610}{96969512287777806715195515011040582870493345720777793567438861617} a^{34} - \frac{952254442585595558057190654465367552160472310381140030036602775}{96969512287777806715195515011040582870493345720777793567438861617} a^{33} + \frac{9000427524898886345944057545667161645242458600532452283097794895}{193939024575555613430391030022081165740986691441555587134877723234} a^{32} - \frac{18709195675041173664341738893751981838561472031303186945909321520}{96969512287777806715195515011040582870493345720777793567438861617} a^{31} + \frac{69413859066306772756798689560973042545573935668170485838931269100}{96969512287777806715195515011040582870493345720777793567438861617} a^{30} - \frac{232184724496490881834907707690682242689982304326106724489085624980}{96969512287777806715195515011040582870493345720777793567438861617} a^{29} + \frac{30679715143774017168094092888493037406115000751715043647392313270}{4216065751642513335443283261349590559586667205251208415975602679} a^{28} - \frac{85200940005550570850807205121537352018338900489140094644375339240}{4216065751642513335443283261349590559586667205251208415975602679} a^{27} + \frac{4994523884744218620301895303030823118383933253660407005964470221980}{96969512287777806715195515011040582870493345720777793567438861617} a^{26} - \frac{11719634330889640607991927913552529407776183440223836135420265368820}{96969512287777806715195515011040582870493345720777793567438861617} a^{25} + \frac{25375077676436008690105392309295738302376668423442259965303985348282}{96969512287777806715195515011040582870493345720777793567438861617} a^{24} - \frac{50770777958717102328896528995269421696802973122193517551518667065184}{96969512287777806715195515011040582870493345720777793567438861617} a^{23} + \frac{4085004519735512952394429493759333403225808005927547408616783795786}{4216065751642513335443283261349590559586667205251208415975602679} a^{22} - \frac{6995154640473396039026660016877742802276149955113691568744746099542}{4216065751642513335443283261349590559586667205251208415975602679} a^{21} + \frac{11087460052230055671095732578012365440266301365000692788571904633519}{4216065751642513335443283261349590559586667205251208415975602679} a^{20} - \frac{16274144928807593905612695568397579492160016954548741491224945602548}{4216065751642513335443283261349590559586667205251208415975602679} a^{19} + \frac{22146461695800234917300233992972743486256279353402467156902737349126}{4216065751642513335443283261349590559586667205251208415975602679} a^{18} - \frac{28010536150937654430162927356593495145327889995943649485220373634192}{4216065751642513335443283261349590559586667205251208415975602679} a^{17} + \frac{760455948701635293639139826283856977228334230785137147039545841929692}{96969512287777806715195515011040582870493345720777793567438861617} a^{16} - \frac{842199190329141675405573618934073543404578758188901487910593306743184}{96969512287777806715195515011040582870493345720777793567438861617} a^{15} + \frac{877827306918883358224698292420744439910847807177128627770719870521928}{96969512287777806715195515011040582870493345720777793567438861617} a^{14} - \frac{856416085009785779306800405338308526969204417433912776068173488597376}{96969512287777806715195515011040582870493345720777793567438861617} a^{13} + \frac{762999578334689248439377356015126339472940786776955953970564291253872}{96969512287777806715195515011040582870493345720777793567438861617} a^{12} - \frac{585311118074001816285396738889471603178545201015008072042111213790144}{96969512287777806715195515011040582870493345720777793567438861617} a^{11} + \frac{380941114221563055029551878707849208358103031418329752537266680380448}{96969512287777806715195515011040582870493345720777793567438861617} a^{10} - \frac{295256581183384722912558339946109959743904504782638442833497262896896}{96969512287777806715195515011040582870493345720777793567438861617} a^{9} + \frac{318952686561887837766903402161859451557199578488868283256433751136192}{96969512287777806715195515011040582870493345720777793567438861617} a^{8} - \frac{235959032667300740916413344231610517992847338178508680939723691762944}{96969512287777806715195515011040582870493345720777793567438861617} a^{7} + \frac{2115023443667562235546037960102072688274914892452790526005398759296}{4216065751642513335443283261349590559586667205251208415975602679} a^{6} + \frac{1395623727336515195815806936019026131610477882219984070234736386048}{4216065751642513335443283261349590559586667205251208415975602679} a^{5} - \frac{2780056187717305487724500388166859713316716926487802218305293003008}{96969512287777806715195515011040582870493345720777793567438861617} a^{4} - \frac{15794979047191414193304232521883266685784185402301348433772426415104}{96969512287777806715195515011040582870493345720777793567438861617} a^{3} + \frac{13836534115153001991692990196134657239350001406176267515173337607887}{193939024575555613430391030022081165740986691441555587134877723234} a^{2} - \frac{469246075994618134753986003585689648856601890114751787470310925114}{96969512287777806715195515011040582870493345720777793567438861617} a - \frac{1148884082162065052949607959839741037932899797609469154890297758}{4216065751642513335443283261349590559586667205251208415975602679} \) (15041490416277220308715037744496914119905014927547888601830)/(96969512287777806715195515011040582870493345720777793567438861617)a^(38) - (285788317909267185865585717145441368278195283623409883434770)/(96969512287777806715195515011040582870493345720777793567438861617)a^(37) + (3236868595667634317862414339534700504295561515362787074414765)/(96969512287777806715195515011040582870493345720777793567438861617)a^(36) - (26541131434088760090443443508480617198440430794331670278206300)/(96969512287777806715195515011040582870493345720777793567438861617)a^(35) + (173828646740518385726063441927098406984786359642476106328063610)/(96969512287777806715195515011040582870493345720777793567438861617)a^(34) - (952254442585595558057190654465367552160472310381140030036602775)/(96969512287777806715195515011040582870493345720777793567438861617)a^(33) + (9000427524898886345944057545667161645242458600532452283097794895)/(193939024575555613430391030022081165740986691441555587134877723234)a^(32) - (18709195675041173664341738893751981838561472031303186945909321520)/(96969512287777806715195515011040582870493345720777793567438861617)a^(31) + (69413859066306772756798689560973042545573935668170485838931269100)/(96969512287777806715195515011040582870493345720777793567438861617)a^(30) - (232184724496490881834907707690682242689982304326106724489085624980)/(96969512287777806715195515011040582870493345720777793567438861617)a^(29) + (30679715143774017168094092888493037406115000751715043647392313270)/(4216065751642513335443283261349590559586667205251208415975602679)a^(28) - (85200940005550570850807205121537352018338900489140094644375339240)/(4216065751642513335443283261349590559586667205251208415975602679)a^(27) + (4994523884744218620301895303030823118383933253660407005964470221980)/(96969512287777806715195515011040582870493345720777793567438861617)a^(26) - (11719634330889640607991927913552529407776183440223836135420265368820)/(96969512287777806715195515011040582870493345720777793567438861617)a^(25) + (25375077676436008690105392309295738302376668423442259965303985348282)/(96969512287777806715195515011040582870493345720777793567438861617)a^(24) - (50770777958717102328896528995269421696802973122193517551518667065184)/(96969512287777806715195515011040582870493345720777793567438861617)a^(23) + (4085004519735512952394429493759333403225808005927547408616783795786)/(4216065751642513335443283261349590559586667205251208415975602679)a^(22) - (6995154640473396039026660016877742802276149955113691568744746099542)/(4216065751642513335443283261349590559586667205251208415975602679)a^(21) + (11087460052230055671095732578012365440266301365000692788571904633519)/(4216065751642513335443283261349590559586667205251208415975602679)a^(20) - (16274144928807593905612695568397579492160016954548741491224945602548)/(4216065751642513335443283261349590559586667205251208415975602679)a^(19) + (22146461695800234917300233992972743486256279353402467156902737349126)/(4216065751642513335443283261349590559586667205251208415975602679)a^(18) - (28010536150937654430162927356593495145327889995943649485220373634192)/(4216065751642513335443283261349590559586667205251208415975602679)a^(17) + (760455948701635293639139826283856977228334230785137147039545841929692)/(96969512287777806715195515011040582870493345720777793567438861617)a^(16) - (842199190329141675405573618934073543404578758188901487910593306743184)/(96969512287777806715195515011040582870493345720777793567438861617)a^(15) + (877827306918883358224698292420744439910847807177128627770719870521928)/(96969512287777806715195515011040582870493345720777793567438861617)a^(14) - (856416085009785779306800405338308526969204417433912776068173488597376)/(96969512287777806715195515011040582870493345720777793567438861617)a^(13) + (762999578334689248439377356015126339472940786776955953970564291253872)/(96969512287777806715195515011040582870493345720777793567438861617)a^(12) - (585311118074001816285396738889471603178545201015008072042111213790144)/(96969512287777806715195515011040582870493345720777793567438861617)a^(11) + (380941114221563055029551878707849208358103031418329752537266680380448)/(96969512287777806715195515011040582870493345720777793567438861617)a^(10) - (295256581183384722912558339946109959743904504782638442833497262896896)/(96969512287777806715195515011040582870493345720777793567438861617)a^(9) + (318952686561887837766903402161859451557199578488868283256433751136192)/(96969512287777806715195515011040582870493345720777793567438861617)a^(8) - (235959032667300740916413344231610517992847338178508680939723691762944)/(96969512287777806715195515011040582870493345720777793567438861617)a^(7) + (2115023443667562235546037960102072688274914892452790526005398759296)/(4216065751642513335443283261349590559586667205251208415975602679)a^(6) + (1395623727336515195815806936019026131610477882219984070234736386048)/(4216065751642513335443283261349590559586667205251208415975602679)a^(5) - (2780056187717305487724500388166859713316716926487802218305293003008)/(96969512287777806715195515011040582870493345720777793567438861617)a^(4) - (15794979047191414193304232521883266685784185402301348433772426415104)/(96969512287777806715195515011040582870493345720777793567438861617)a^(3) + (13836534115153001991692990196134657239350001406176267515173337607887)/(193939024575555613430391030022081165740986691441555587134877723234)a^(2) - (469246075994618134753986003585689648856601890114751787470310925114)/(96969512287777806715195515011040582870493345720777793567438861617)a - (1148884082162065052949607959839741037932899797609469154890297758)/(4216065751642513335443283261349590559586667205251208415975602679) (order $22$)	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^40 - 20*x^39 + 234*x^38 - 1976*x^37 + 13279*x^36 - 74520*x^35 + 360238*x^34 - 1530680*x^33 + 5800422*x^32 - 19809432*x^31 + 61453956*x^30 - 174207264*x^29 + 453277478*x^28 - 1086128128*x^27 + 2402433828*x^26 - 4913327328*x^25 + 9300148725*x^24 - 16301626740*x^23 + 26468749370*x^22 - 39824889160*x^21 + 55573165179*x^20 - 72059216200*x^19 + 87116562190*x^18 - 98665586280*x^17 + 105104472084*x^16 - 105116990208*x^15 + 97114298392*x^14 - 79416402656*x^13 + 55222821968*x^12 - 38173429376*x^11 + 36542373728*x^10 - 33447633536*x^9 + 14972299072*x^8 + 1667546112*x^7 - 2576594560*x^6 - 1340352000*x^5 + 1398502656*x^4 - 272152576*x^3 - 20025856*x^2 + 5699584*x + 541696)

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^40 - 20*x^39 + 234*x^38 - 1976*x^37 + 13279*x^36 - 74520*x^35 + 360238*x^34 - 1530680*x^33 + 5800422*x^32 - 19809432*x^31 + 61453956*x^30 - 174207264*x^29 + 453277478*x^28 - 1086128128*x^27 + 2402433828*x^26 - 4913327328*x^25 + 9300148725*x^24 - 16301626740*x^23 + 26468749370*x^22 - 39824889160*x^21 + 55573165179*x^20 - 72059216200*x^19 + 87116562190*x^18 - 98665586280*x^17 + 105104472084*x^16 - 105116990208*x^15 + 97114298392*x^14 - 79416402656*x^13 + 55222821968*x^12 - 38173429376*x^11 + 36542373728*x^10 - 33447633536*x^9 + 14972299072*x^8 + 1667546112*x^7 - 2576594560*x^6 - 1340352000*x^5 + 1398502656*x^4 - 272152576*x^3 - 20025856*x^2 + 5699584*x + 541696, 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^40 - 20*x^39 + 234*x^38 - 1976*x^37 + 13279*x^36 - 74520*x^35 + 360238*x^34 - 1530680*x^33 + 5800422*x^32 - 19809432*x^31 + 61453956*x^30 - 174207264*x^29 + 453277478*x^28 - 1086128128*x^27 + 2402433828*x^26 - 4913327328*x^25 + 9300148725*x^24 - 16301626740*x^23 + 26468749370*x^22 - 39824889160*x^21 + 55573165179*x^20 - 72059216200*x^19 + 87116562190*x^18 - 98665586280*x^17 + 105104472084*x^16 - 105116990208*x^15 + 97114298392*x^14 - 79416402656*x^13 + 55222821968*x^12 - 38173429376*x^11 + 36542373728*x^10 - 33447633536*x^9 + 14972299072*x^8 + 1667546112*x^7 - 2576594560*x^6 - 1340352000*x^5 + 1398502656*x^4 - 272152576*x^3 - 20025856*x^2 + 5699584*x + 541696);

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^40 - 20*x^39 + 234*x^38 - 1976*x^37 + 13279*x^36 - 74520*x^35 + 360238*x^34 - 1530680*x^33 + 5800422*x^32 - 19809432*x^31 + 61453956*x^30 - 174207264*x^29 + 453277478*x^28 - 1086128128*x^27 + 2402433828*x^26 - 4913327328*x^25 + 9300148725*x^24 - 16301626740*x^23 + 26468749370*x^22 - 39824889160*x^21 + 55573165179*x^20 - 72059216200*x^19 + 87116562190*x^18 - 98665586280*x^17 + 105104472084*x^16 - 105116990208*x^15 + 97114298392*x^14 - 79416402656*x^13 + 55222821968*x^12 - 38173429376*x^11 + 36542373728*x^10 - 33447633536*x^9 + 14972299072*x^8 + 1667546112*x^7 - 2576594560*x^6 - 1340352000*x^5 + 1398502656*x^4 - 272152576*x^3 - 20025856*x^2 + 5699584*x + 541696);

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

$C_2^2\times C_{10}$ (as 40T7):

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)

An abelian group of order 40

The 40 conjugacy class representatives for $C_2^2\times C_{10}$

Character table for $C_2^2\times C_{10}$

Intermediate fields

$\Q(\sqrt{77}) $, $\Q(\sqrt{-22}) $, $\Q(\sqrt{-14}) $, $\Q(\sqrt{-11}) $, $\Q(\sqrt{-7}) $, $\Q(\sqrt{2}) $, $\Q(\sqrt{154}) $, $\Q(\sqrt{-14}, \sqrt{-22})$, $\Q(\sqrt{-7}, \sqrt{-11})$, $\Q(\sqrt{2}, \sqrt{77})$, $\Q(\sqrt{2}, \sqrt{-11})$, $\Q(\sqrt{-7}, \sqrt{-22})$, $\Q(\sqrt{-11}, \sqrt{-14})$, $\Q(\sqrt{2}, \sqrt{-7})$, $\Q(\zeta_{11})^+$, 8.0.143986855936.2, 10.10.39630026842637.1, 10.0.77265229938688.1, 10.0.118054247234502656.1, $\Q(\zeta_{11})$, 10.0.3602729712967.1, 10.10.7024111812608.1, 10.10.1298596719579529216.1, 20.0.1686353440102714438260338720197574656.2, 20.0.1570539027548129147161113769.2, 20.20.1686353440102714438260338720197574656.1, 20.0.5969915757478328440239161344.6, 20.0.1686353440102714438260338720197574656.1, 20.0.1686353440102714438260338720197574656.8, 20.0.13936805290105078002151559671054336.6

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]

Frobenius cycle types

$p$	$2$	$3$	$5$	$7$	$11$	$13$	$17$	$19$	$23$	$29$	$31$	$37$	$41$	$43$	$47$	$53$	$59$
Cycle type	R	${\href{/padicField/3.10.0.1}{10} }^{4}$	${\href{/padicField/5.10.0.1}{10} }^{4}$	R	R	${\href{/padicField/13.10.0.1}{10} }^{4}$	${\href{/padicField/17.10.0.1}{10} }^{4}$	${\href{/padicField/19.10.0.1}{10} }^{4}$	${\href{/padicField/23.1.0.1}{1} }^{40}$	${\href{/padicField/29.10.0.1}{10} }^{4}$	${\href{/padicField/31.10.0.1}{10} }^{4}$	${\href{/padicField/37.10.0.1}{10} }^{4}$	${\href{/padicField/41.10.0.1}{10} }^{4}$	${\href{/padicField/43.2.0.1}{2} }^{20}$	${\href{/padicField/47.10.0.1}{10} }^{4}$	${\href{/padicField/53.10.0.1}{10} }^{4}$	${\href{/padicField/59.10.0.1}{10} }^{4}$

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		Deg $20$	$2$	$10$	$30$
$2$ 2		Deg $20$	$2$	$10$	$30$
$7$ 7	7.20.10.1	$x^{20} + 70 x^{18} + 2207 x^{16} + 2 x^{15} + 41168 x^{14} - 68 x^{13} + 501639 x^{12} - 3674 x^{11} + 4175501 x^{10} - 48430 x^{9} + 24202032 x^{8} - 163712 x^{7} + 97377995 x^{6} + 430996 x^{5} + 259701777 x^{4} + 2947158 x^{3} + 412861211 x^{2} + 7541370 x + 287825400$	$2$	$10$	$10$	20T3	$[\ ]_{2}^{10}$
$7$ 7	7.20.10.1	$x^{20} + 70 x^{18} + 2207 x^{16} + 2 x^{15} + 41168 x^{14} - 68 x^{13} + 501639 x^{12} - 3674 x^{11} + 4175501 x^{10} - 48430 x^{9} + 24202032 x^{8} - 163712 x^{7} + 97377995 x^{6} + 430996 x^{5} + 259701777 x^{4} + 2947158 x^{3} + 412861211 x^{2} + 7541370 x + 287825400$	$2$	$10$	$10$	20T3	$[\ ]_{2}^{10}$
$11$ 11	11.20.18.1	$x^{20} + 70 x^{19} + 2225 x^{18} + 42420 x^{17} + 539670 x^{16} + 4821684 x^{15} + 31004730 x^{14} + 144683280 x^{13} + 488310165 x^{12} + 1177567510 x^{11} + 1996241675 x^{10} + 2355135790 x^{9} + 1953262935 x^{8} + 1157863560 x^{7} + 500734950 x^{6} + 191763012 x^{5} + 243790230 x^{4} + 806750280 x^{3} + 2014356815 x^{2} + 2999040310 x + 2009802620$	$10$	$2$	$18$	20T3	$[\ ]_{10}^{2}$
$11$ 11	11.20.18.1	$x^{20} + 70 x^{19} + 2225 x^{18} + 42420 x^{17} + 539670 x^{16} + 4821684 x^{15} + 31004730 x^{14} + 144683280 x^{13} + 488310165 x^{12} + 1177567510 x^{11} + 1996241675 x^{10} + 2355135790 x^{9} + 1953262935 x^{8} + 1157863560 x^{7} + 500734950 x^{6} + 191763012 x^{5} + 243790230 x^{4} + 806750280 x^{3} + 2014356815 x^{2} + 2999040310 x + 2009802620$	$10$	$2$	$18$	20T3	$[\ ]_{10}^{2}$

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