LMFDB - Number field 24.0.1704017867935867218637163902534560973508837376.2

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^24 + 78*x^22 + 2421*x^20 + 39474*x^18 + 376092*x^16 + 2183742*x^14 + 7748525*x^12 + 16261674*x^10 + 18591153*x^8 + 9415392*x^6 + 875496*x^4 + 15840*x^2 + 16)

gp: K = bnfinit(y^24 + 78*y^22 + 2421*y^20 + 39474*y^18 + 376092*y^16 + 2183742*y^14 + 7748525*y^12 + 16261674*y^10 + 18591153*y^8 + 9415392*y^6 + 875496*y^4 + 15840*y^2 + 16, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^24 + 78*x^22 + 2421*x^20 + 39474*x^18 + 376092*x^16 + 2183742*x^14 + 7748525*x^12 + 16261674*x^10 + 18591153*x^8 + 9415392*x^6 + 875496*x^4 + 15840*x^2 + 16);

oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^24 + 78*x^22 + 2421*x^20 + 39474*x^18 + 376092*x^16 + 2183742*x^14 + 7748525*x^12 + 16261674*x^10 + 18591153*x^8 + 9415392*x^6 + 875496*x^4 + 15840*x^2 + 16)

$ x^{24} + 78 x^{22} + 2421 x^{20} + 39474 x^{18} + 376092 x^{16} + 2183742 x^{14} + 7748525 x^{12} + \cdots + 16 $ x^24 + 78*x^22 + 2421*x^20 + 39474*x^18 + 376092*x^16 + 2183742*x^14 + 7748525*x^12 + 16261674*x^10 + 18591153*x^8 + 9415392*x^6 + 875496*x^4 + 15840*x^2 + 16

sage: K.defining_polynomial()

gp: K.pol

magma: DefiningPolynomial(K);

oscar: defining_polynomial(K)

Invariants

Degree:	$24$	sage: K.degree() gp: poldegree(K.pol) magma: Degree(K); oscar: degree(K)
Signature:	$[0, 12]$	sage: K.signature() gp: K.sign magma: Signature(K); oscar: signature(K)
Discriminant:	$1704017867935867218637163902534560973508837376$ 1704017867935867218637163902534560973508837376 $\medspace = 2^{36}\cdot 3^{28}\cdot 7^{12}\cdot 23^{8}$	sage: K.disc() gp: K.disc magma: OK := Integers(K); Discriminant(OK); oscar: OK = ring_of_integers(K); discriminant(OK)
Root discriminant:	$76.67$	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:	$2^{3/2}3^{7/6}7^{1/2}23^{1/2}\approx 129.30025916061993$
Ramified primes:	$2$, $3$, $7$, $23$ 2, 3, 7, 23	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$^{2048}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{4}a^{8}-\frac{1}{2}a^{5}+\frac{1}{4}a^{2}-\frac{1}{2}$, $\frac{1}{4}a^{9}-\frac{1}{2}a^{6}+\frac{1}{4}a^{3}-\frac{1}{2}a$, $\frac{1}{4}a^{10}-\frac{1}{2}a^{7}+\frac{1}{4}a^{4}-\frac{1}{2}a^{2}$, $\frac{1}{4}a^{11}+\frac{1}{4}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}$, $\frac{1}{4}a^{12}+\frac{1}{4}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}$, $\frac{1}{4}a^{13}+\frac{1}{4}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}$, $\frac{1}{4}a^{14}-\frac{1}{2}a^{6}-\frac{1}{4}a^{2}-\frac{1}{2}$, $\frac{1}{4}a^{15}-\frac{1}{2}a^{7}-\frac{1}{4}a^{3}-\frac{1}{2}a$, $\frac{1}{32}a^{16}-\frac{1}{8}a^{15}-\frac{1}{8}a^{14}-\frac{1}{16}a^{10}-\frac{1}{8}a^{8}-\frac{1}{4}a^{7}-\frac{1}{4}a^{6}-\frac{1}{2}a^{5}-\frac{15}{32}a^{4}-\frac{3}{8}a^{3}-\frac{1}{2}a^{2}-\frac{1}{4}a+\frac{3}{8}$, $\frac{1}{64}a^{17}+\frac{1}{16}a^{15}-\frac{1}{8}a^{14}+\frac{3}{32}a^{11}+\frac{1}{16}a^{9}+\frac{3}{8}a^{7}+\frac{1}{4}a^{6}+\frac{9}{64}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}+\frac{1}{8}a^{2}+\frac{3}{16}a-\frac{1}{4}$, $\frac{1}{128}a^{18}-\frac{1}{64}a^{16}-\frac{1}{16}a^{14}-\frac{5}{64}a^{12}-\frac{1}{8}a^{8}-\frac{1}{2}a^{7}-\frac{23}{128}a^{6}+\frac{5}{64}a^{4}-\frac{1}{2}a^{3}-\frac{5}{32}a^{2}-\frac{1}{2}a-\frac{1}{16}$, $\frac{1}{128}a^{19}-\frac{1}{8}a^{14}-\frac{5}{64}a^{13}+\frac{3}{32}a^{11}-\frac{1}{16}a^{9}+\frac{25}{128}a^{7}+\frac{1}{4}a^{6}+\frac{7}{32}a^{5}+\frac{11}{32}a^{3}+\frac{1}{8}a^{2}+\frac{1}{8}a-\frac{1}{4}$, $\frac{1}{17210624}a^{20}-\frac{1}{256}a^{19}+\frac{73}{108928}a^{18}-\frac{1}{128}a^{17}+\frac{4537}{2151328}a^{16}-\frac{1}{32}a^{15}+\frac{18141}{374144}a^{14}-\frac{11}{128}a^{13}-\frac{8713}{537832}a^{12}-\frac{3}{32}a^{11}+\frac{55997}{1075664}a^{10}+\frac{1824649}{17210624}a^{8}+\frac{23}{256}a^{7}+\frac{54477}{8605312}a^{6}-\frac{55}{128}a^{5}+\frac{2079439}{4302656}a^{4}-\frac{27}{64}a^{3}-\frac{437901}{2151328}a^{2}-\frac{5}{32}a-\frac{52485}{134458}$, $\frac{1}{17210624}a^{21}-\frac{705}{217856}a^{19}-\frac{49081}{8605312}a^{17}-\frac{40319}{374144}a^{15}-\frac{1}{8}a^{14}-\frac{878927}{8605312}a^{13}-\frac{89693}{2151328}a^{11}+\frac{1824649}{17210624}a^{9}-\frac{2647435}{17210624}a^{7}-\frac{1}{4}a^{6}+\frac{461283}{8605312}a^{5}+\frac{2149503}{4302656}a^{3}-\frac{3}{8}a^{2}+\frac{437591}{2151328}a+\frac{1}{4}$, $\frac{1}{67\!\cdots\!12}a^{22}+\frac{907922540371}{67\!\cdots\!12}a^{20}-\frac{1}{256}a^{19}+\frac{29\!\cdots\!37}{16\!\cdots\!28}a^{18}-\frac{1}{128}a^{17}-\frac{47\!\cdots\!43}{33\!\cdots\!56}a^{16}-\frac{1}{32}a^{15}+\frac{36\!\cdots\!07}{33\!\cdots\!56}a^{14}-\frac{11}{128}a^{13}-\frac{81\!\cdots\!39}{16\!\cdots\!28}a^{12}-\frac{3}{32}a^{11}+\frac{25\!\cdots\!53}{67\!\cdots\!12}a^{10}-\frac{59\!\cdots\!85}{67\!\cdots\!12}a^{8}+\frac{23}{256}a^{7}+\frac{13\!\cdots\!53}{16\!\cdots\!28}a^{6}-\frac{55}{128}a^{5}-\frac{31\!\cdots\!45}{21\!\cdots\!16}a^{4}+\frac{5}{64}a^{3}+\frac{77\!\cdots\!79}{21\!\cdots\!16}a^{2}-\frac{5}{32}a+\frac{50\!\cdots\!57}{42\!\cdots\!32}$, $\frac{1}{13\!\cdots\!24}a^{23}-\frac{1}{13\!\cdots\!24}a^{22}-\frac{3012355647067}{13\!\cdots\!24}a^{21}+\frac{3012355647067}{13\!\cdots\!24}a^{20}-\frac{11\!\cdots\!87}{33\!\cdots\!56}a^{19}-\frac{571852107486827}{10\!\cdots\!08}a^{18}-\frac{19\!\cdots\!63}{67\!\cdots\!12}a^{17}+\frac{54\!\cdots\!67}{67\!\cdots\!12}a^{16}+\frac{40\!\cdots\!89}{67\!\cdots\!12}a^{15}+\frac{64\!\cdots\!91}{67\!\cdots\!12}a^{14}+\frac{15\!\cdots\!23}{33\!\cdots\!56}a^{13}+\frac{40\!\cdots\!19}{33\!\cdots\!56}a^{12}-\frac{99\!\cdots\!23}{13\!\cdots\!24}a^{11}+\frac{99\!\cdots\!23}{13\!\cdots\!24}a^{10}-\frac{46\!\cdots\!83}{13\!\cdots\!24}a^{9}+\frac{13\!\cdots\!47}{13\!\cdots\!24}a^{8}+\frac{42\!\cdots\!63}{33\!\cdots\!56}a^{7}-\frac{27\!\cdots\!11}{16\!\cdots\!28}a^{6}+\frac{60\!\cdots\!47}{42\!\cdots\!32}a^{5}+\frac{11\!\cdots\!89}{16\!\cdots\!28}a^{4}+\frac{38\!\cdots\!79}{10\!\cdots\!08}a^{3}+\frac{74\!\cdots\!47}{84\!\cdots\!64}a^{2}-\frac{18\!\cdots\!33}{84\!\cdots\!64}a+\frac{41\!\cdots\!51}{84\!\cdots\!64}$ 1, a, a^2, a^3, a^4, a^5, a^6, a^7, 1/4*a^8 - 1/2*a^5 + 1/4*a^2 - 1/2, 1/4*a^9 - 1/2*a^6 + 1/4*a^3 - 1/2*a, 1/4*a^10 - 1/2*a^7 + 1/4*a^4 - 1/2*a^2, 1/4*a^11 + 1/4*a^5 - 1/2*a^3 - 1/2*a^2, 1/4*a^12 + 1/4*a^6 - 1/2*a^4 - 1/2*a^3, 1/4*a^13 + 1/4*a^7 - 1/2*a^5 - 1/2*a^4, 1/4*a^14 - 1/2*a^6 - 1/4*a^2 - 1/2, 1/4*a^15 - 1/2*a^7 - 1/4*a^3 - 1/2*a, 1/32*a^16 - 1/8*a^15 - 1/8*a^14 - 1/16*a^10 - 1/8*a^8 - 1/4*a^7 - 1/4*a^6 - 1/2*a^5 - 15/32*a^4 - 3/8*a^3 - 1/2*a^2 - 1/4*a + 3/8, 1/64*a^17 + 1/16*a^15 - 1/8*a^14 + 3/32*a^11 + 1/16*a^9 + 3/8*a^7 + 1/4*a^6 + 9/64*a^5 - 1/2*a^4 - 1/2*a^3 + 1/8*a^2 + 3/16*a - 1/4, 1/128*a^18 - 1/64*a^16 - 1/16*a^14 - 5/64*a^12 - 1/8*a^8 - 1/2*a^7 - 23/128*a^6 + 5/64*a^4 - 1/2*a^3 - 5/32*a^2 - 1/2*a - 1/16, 1/128*a^19 - 1/8*a^14 - 5/64*a^13 + 3/32*a^11 - 1/16*a^9 + 25/128*a^7 + 1/4*a^6 + 7/32*a^5 + 11/32*a^3 + 1/8*a^2 + 1/8*a - 1/4, 1/17210624*a^20 - 1/256*a^19 + 73/108928*a^18 - 1/128*a^17 + 4537/2151328*a^16 - 1/32*a^15 + 18141/374144*a^14 - 11/128*a^13 - 8713/537832*a^12 - 3/32*a^11 + 55997/1075664*a^10 + 1824649/17210624*a^8 + 23/256*a^7 + 54477/8605312*a^6 - 55/128*a^5 + 2079439/4302656*a^4 - 27/64*a^3 - 437901/2151328*a^2 - 5/32*a - 52485/134458, 1/17210624*a^21 - 705/217856*a^19 - 49081/8605312*a^17 - 40319/374144*a^15 - 1/8*a^14 - 878927/8605312*a^13 - 89693/2151328*a^11 + 1824649/17210624*a^9 - 2647435/17210624*a^7 - 1/4*a^6 + 461283/8605312*a^5 + 2149503/4302656*a^3 - 3/8*a^2 + 437591/2151328*a + 1/4, 1/67470433859396941312*a^22 + 907922540371/67470433859396941312*a^20 - 1/256*a^19 + 29603389593055937/16867608464849235328*a^18 - 1/128*a^17 - 475119244815043943/33735216929698470656*a^16 - 1/32*a^15 + 3616438152058578607/33735216929698470656*a^14 - 11/128*a^13 - 81067445941457339/16867608464849235328*a^12 - 3/32*a^11 + 2515970372216083353/67470433859396941312*a^10 - 5965306154630783385/67470433859396941312*a^8 + 23/256*a^7 + 1309116321798439353/16867608464849235328*a^6 - 55/128*a^5 - 319100316645775045/2108451058106154416*a^4 + 5/64*a^3 + 779450825924012579/2108451058106154416*a^2 - 5/32*a + 504114689706627457/4216902116212308832, 1/134940867718793882624*a^23 - 1/134940867718793882624*a^22 - 3012355647067/134940867718793882624*a^21 + 3012355647067/134940867718793882624*a^20 - 113478923692056187/33735216929698470656*a^19 - 571852107486827/1054225529053077208*a^18 - 19151688834130163/67470433859396941312*a^17 + 546264453360668767/67470433859396941312*a^16 + 4089180578403539589/67470433859396941312*a^15 + 6453074712127232491/67470433859396941312*a^14 + 1509973536152067523/33735216929698470656*a^13 + 4024710491376587819/33735216929698470656*a^12 - 996410710375367623/134940867718793882624*a^11 + 996410710375367623/134940867718793882624*a^10 - 4684633596636724983/134940867718793882624*a^9 + 13118437829061342647/134940867718793882624*a^8 + 4233232220417506363/33735216929698470656*a^7 - 2709617970301109111/16867608464849235328*a^6 + 605630583319850147/4216902116212308832*a^5 + 1135488827274734989/16867608464849235328*a^4 + 384517434598105279/1054225529053077208*a^3 + 745428066032562647/8433804232424617664*a^2 - 1803184638900944633/8433804232424617664*a + 4175192079270368351/8433804232424617664

sage: K.integral_basis()

gp: K.zk

magma: IntegralBasis(K);

oscar: basis(OK)

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

Class group and class number

$C_{2}\times C_{4}\times C_{240}$, which has order $1920$ (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:	$11$	sage: UK.rank() gp: K.fu magma: UnitRank(K); oscar: rank(UK)
Torsion generator:	$ -\frac{16977928905}{7138430691584} a^{23} - \frac{41382739415}{223075959112} a^{21} - \frac{41101133468325}{7138430691584} a^{19} - \frac{167527454174045}{1784607672896} a^{17} - \frac{1595993229742529}{1784607672896} a^{15} - \frac{18531310168497875}{3569215345792} a^{13} - \frac{131475329590063473}{7138430691584} a^{11} - \frac{68947123119872755}{1784607672896} a^{9} - \frac{13693015972127631}{310366551808} a^{7} - \frac{79471318513897261}{3569215345792} a^{5} - \frac{3580909509496679}{1784607672896} a^{3} - \frac{25489141306319}{892303836448} a + \frac{1}{2} $ -(16977928905)/(7138430691584)a^(23) - (41382739415)/(223075959112)a^(21) - (41101133468325)/(7138430691584)a^(19) - (167527454174045)/(1784607672896)a^(17) - (1595993229742529)/(1784607672896)a^(15) - (18531310168497875)/(3569215345792)a^(13) - (131475329590063473)/(7138430691584)a^(11) - (68947123119872755)/(1784607672896)a^(9) - (13693015972127631)/(310366551808)a^(7) - (79471318513897261)/(3569215345792)a^(5) - (3580909509496679)/(1784607672896)a^(3) - (25489141306319)/(892303836448)a + (1)/(2) (order $6$)	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:	$\frac{423404988277887}{85\!\cdots\!28}a^{23}+\frac{32\!\cdots\!79}{85\!\cdots\!28}a^{21}+\frac{12\!\cdots\!15}{10\!\cdots\!16}a^{19}+\frac{83\!\cdots\!11}{42\!\cdots\!64}a^{17}+\frac{79\!\cdots\!11}{42\!\cdots\!64}a^{15}+\frac{22\!\cdots\!91}{21\!\cdots\!32}a^{13}+\frac{32\!\cdots\!27}{85\!\cdots\!28}a^{11}+\frac{66\!\cdots\!27}{85\!\cdots\!28}a^{9}+\frac{20\!\cdots\!29}{23\!\cdots\!96}a^{7}+\frac{43\!\cdots\!73}{10\!\cdots\!16}a^{5}+\frac{64\!\cdots\!05}{53\!\cdots\!08}a^{3}-\frac{73\!\cdots\!89}{53\!\cdots\!08}a$, $\frac{41\!\cdots\!15}{33\!\cdots\!56}a^{23}-\frac{3940387}{22381668224}a^{22}+\frac{81\!\cdots\!15}{84\!\cdots\!64}a^{21}-\frac{153630661}{11190834112}a^{20}+\frac{10\!\cdots\!19}{33\!\cdots\!56}a^{19}-\frac{9532921237}{22381668224}a^{18}+\frac{41\!\cdots\!71}{84\!\cdots\!64}a^{17}-\frac{19417655015}{2797708528}a^{16}+\frac{39\!\cdots\!05}{84\!\cdots\!64}a^{15}-\frac{184856597939}{2797708528}a^{14}+\frac{45\!\cdots\!17}{16\!\cdots\!28}a^{13}-\frac{4288945569495}{11190834112}a^{12}+\frac{32\!\cdots\!59}{33\!\cdots\!56}a^{11}-\frac{30397069709563}{22381668224}a^{10}+\frac{52\!\cdots\!05}{26\!\cdots\!02}a^{9}-\frac{31849594502383}{11190834112}a^{8}+\frac{33\!\cdots\!09}{14\!\cdots\!72}a^{7}-\frac{72728437018777}{22381668224}a^{6}+\frac{19\!\cdots\!99}{16\!\cdots\!28}a^{5}-\frac{18402326664749}{11190834112}a^{4}+\frac{82\!\cdots\!49}{84\!\cdots\!64}a^{3}-\frac{847601185687}{5595417056}a^{2}+\frac{24\!\cdots\!61}{42\!\cdots\!32}a+\frac{320629461}{2797708528}$, $\frac{19654750614585}{45\!\cdots\!88}a^{22}+\frac{15\!\cdots\!57}{45\!\cdots\!88}a^{20}+\frac{185761494720595}{17\!\cdots\!98}a^{18}+\frac{38\!\cdots\!69}{22\!\cdots\!44}a^{16}+\frac{36\!\cdots\!45}{22\!\cdots\!44}a^{14}+\frac{10\!\cdots\!33}{11\!\cdots\!72}a^{12}+\frac{15\!\cdots\!81}{45\!\cdots\!88}a^{10}+\frac{31\!\cdots\!93}{45\!\cdots\!88}a^{8}+\frac{43\!\cdots\!19}{56\!\cdots\!36}a^{6}+\frac{20\!\cdots\!59}{56\!\cdots\!36}a^{4}+\frac{22\!\cdots\!61}{28\!\cdots\!68}a^{2}-\frac{38\!\cdots\!83}{28\!\cdots\!68}$, $\frac{82\!\cdots\!25}{16\!\cdots\!28}a^{23}+\frac{13538546485901}{45\!\cdots\!88}a^{22}+\frac{64\!\cdots\!61}{16\!\cdots\!28}a^{21}+\frac{10\!\cdots\!79}{45\!\cdots\!88}a^{20}+\frac{39\!\cdots\!97}{33\!\cdots\!56}a^{19}+\frac{82\!\cdots\!11}{11\!\cdots\!72}a^{18}+\frac{32\!\cdots\!01}{16\!\cdots\!28}a^{17}+\frac{27\!\cdots\!45}{22\!\cdots\!44}a^{16}+\frac{15\!\cdots\!47}{84\!\cdots\!64}a^{15}+\frac{26\!\cdots\!91}{22\!\cdots\!44}a^{14}+\frac{18\!\cdots\!07}{16\!\cdots\!28}a^{13}+\frac{77\!\cdots\!09}{11\!\cdots\!72}a^{12}+\frac{64\!\cdots\!17}{16\!\cdots\!28}a^{11}+\frac{11\!\cdots\!37}{45\!\cdots\!88}a^{10}+\frac{13\!\cdots\!57}{16\!\cdots\!28}a^{9}+\frac{24\!\cdots\!63}{45\!\cdots\!88}a^{8}+\frac{13\!\cdots\!91}{14\!\cdots\!72}a^{7}+\frac{73\!\cdots\!91}{11\!\cdots\!72}a^{6}+\frac{81\!\cdots\!65}{16\!\cdots\!28}a^{5}+\frac{65\!\cdots\!69}{17\!\cdots\!98}a^{4}+\frac{43\!\cdots\!79}{84\!\cdots\!64}a^{3}+\frac{21\!\cdots\!73}{35\!\cdots\!96}a^{2}-\frac{13\!\cdots\!09}{42\!\cdots\!32}a+\frac{38\!\cdots\!45}{28\!\cdots\!68}$, $\frac{29\!\cdots\!03}{84\!\cdots\!64}a^{23}+\frac{13\!\cdots\!85}{84\!\cdots\!64}a^{22}+\frac{23\!\cdots\!93}{84\!\cdots\!64}a^{21}+\frac{12\!\cdots\!93}{10\!\cdots\!08}a^{20}+\frac{71\!\cdots\!37}{84\!\cdots\!64}a^{19}+\frac{32\!\cdots\!91}{84\!\cdots\!64}a^{18}+\frac{11\!\cdots\!61}{84\!\cdots\!64}a^{17}+\frac{26\!\cdots\!01}{42\!\cdots\!32}a^{16}+\frac{55\!\cdots\!79}{42\!\cdots\!32}a^{15}+\frac{12\!\cdots\!01}{21\!\cdots\!16}a^{14}+\frac{32\!\cdots\!17}{42\!\cdots\!32}a^{13}+\frac{15\!\cdots\!73}{42\!\cdots\!32}a^{12}+\frac{22\!\cdots\!73}{84\!\cdots\!64}a^{11}+\frac{10\!\cdots\!97}{84\!\cdots\!64}a^{10}+\frac{48\!\cdots\!41}{84\!\cdots\!64}a^{9}+\frac{58\!\cdots\!83}{21\!\cdots\!16}a^{8}+\frac{55\!\cdots\!45}{84\!\cdots\!64}a^{7}+\frac{27\!\cdots\!27}{84\!\cdots\!64}a^{6}+\frac{27\!\cdots\!39}{84\!\cdots\!64}a^{5}+\frac{45\!\cdots\!83}{26\!\cdots\!02}a^{4}+\frac{64\!\cdots\!85}{21\!\cdots\!16}a^{3}+\frac{33\!\cdots\!23}{21\!\cdots\!16}a^{2}+\frac{10\!\cdots\!57}{21\!\cdots\!16}a+\frac{15\!\cdots\!99}{52\!\cdots\!04}$, $\frac{98\!\cdots\!49}{13\!\cdots\!24}a^{23}+\frac{54\!\cdots\!95}{13\!\cdots\!24}a^{22}+\frac{77\!\cdots\!41}{13\!\cdots\!24}a^{21}+\frac{42\!\cdots\!63}{13\!\cdots\!24}a^{20}+\frac{59\!\cdots\!27}{33\!\cdots\!56}a^{19}+\frac{82\!\cdots\!83}{84\!\cdots\!64}a^{18}+\frac{19\!\cdots\!73}{67\!\cdots\!12}a^{17}+\frac{10\!\cdots\!51}{67\!\cdots\!12}a^{16}+\frac{18\!\cdots\!05}{67\!\cdots\!12}a^{15}+\frac{10\!\cdots\!39}{67\!\cdots\!12}a^{14}+\frac{23\!\cdots\!17}{14\!\cdots\!72}a^{13}+\frac{29\!\cdots\!11}{33\!\cdots\!56}a^{12}+\frac{33\!\cdots\!43}{58\!\cdots\!88}a^{11}+\frac{42\!\cdots\!91}{13\!\cdots\!24}a^{10}+\frac{16\!\cdots\!77}{13\!\cdots\!24}a^{9}+\frac{88\!\cdots\!95}{13\!\cdots\!24}a^{8}+\frac{45\!\cdots\!29}{33\!\cdots\!56}a^{7}+\frac{12\!\cdots\!99}{16\!\cdots\!28}a^{6}+\frac{58\!\cdots\!17}{84\!\cdots\!64}a^{5}+\frac{63\!\cdots\!13}{16\!\cdots\!28}a^{4}+\frac{26\!\cdots\!07}{42\!\cdots\!32}a^{3}+\frac{28\!\cdots\!79}{84\!\cdots\!64}a^{2}+\frac{91\!\cdots\!83}{84\!\cdots\!64}a+\frac{51\!\cdots\!63}{84\!\cdots\!64}$, $\frac{28\!\cdots\!37}{33\!\cdots\!56}a^{23}+\frac{21\!\cdots\!39}{33\!\cdots\!56}a^{21}+\frac{33\!\cdots\!47}{16\!\cdots\!28}a^{19}+\frac{55\!\cdots\!23}{16\!\cdots\!28}a^{17}+\frac{52\!\cdots\!23}{16\!\cdots\!28}a^{15}+\frac{38\!\cdots\!13}{21\!\cdots\!16}a^{13}+\frac{21\!\cdots\!73}{33\!\cdots\!56}a^{11}+\frac{45\!\cdots\!87}{33\!\cdots\!56}a^{9}+\frac{11\!\cdots\!41}{73\!\cdots\!36}a^{7}+\frac{67\!\cdots\!65}{84\!\cdots\!64}a^{5}+\frac{33\!\cdots\!95}{42\!\cdots\!32}a^{3}+\frac{23\!\cdots\!17}{13\!\cdots\!51}a$, $\frac{13\!\cdots\!61}{33\!\cdots\!56}a^{23}-\frac{38\!\cdots\!15}{16\!\cdots\!28}a^{22}+\frac{52\!\cdots\!21}{16\!\cdots\!28}a^{21}-\frac{30\!\cdots\!63}{16\!\cdots\!28}a^{20}+\frac{32\!\cdots\!87}{33\!\cdots\!56}a^{19}-\frac{12\!\cdots\!07}{22\!\cdots\!48}a^{18}+\frac{32\!\cdots\!03}{21\!\cdots\!16}a^{17}-\frac{74\!\cdots\!95}{84\!\cdots\!64}a^{16}+\frac{15\!\cdots\!33}{10\!\cdots\!08}a^{15}-\frac{70\!\cdots\!83}{84\!\cdots\!64}a^{14}+\frac{14\!\cdots\!29}{16\!\cdots\!28}a^{13}-\frac{19\!\cdots\!83}{42\!\cdots\!32}a^{12}+\frac{10\!\cdots\!25}{33\!\cdots\!56}a^{11}-\frac{27\!\cdots\!39}{16\!\cdots\!28}a^{10}+\frac{10\!\cdots\!71}{16\!\cdots\!28}a^{9}-\frac{55\!\cdots\!99}{16\!\cdots\!28}a^{8}+\frac{24\!\cdots\!03}{33\!\cdots\!56}a^{7}-\frac{75\!\cdots\!09}{21\!\cdots\!16}a^{6}+\frac{62\!\cdots\!47}{16\!\cdots\!28}a^{5}-\frac{36\!\cdots\!29}{21\!\cdots\!16}a^{4}+\frac{28\!\cdots\!09}{84\!\cdots\!64}a^{3}-\frac{89\!\cdots\!41}{45\!\cdots\!96}a^{2}+\frac{24\!\cdots\!09}{42\!\cdots\!32}a-\frac{34\!\cdots\!95}{10\!\cdots\!08}$, $\frac{25664562160333}{56\!\cdots\!36}a^{22}+\frac{250388646982603}{70\!\cdots\!92}a^{20}+\frac{31\!\cdots\!11}{28\!\cdots\!68}a^{18}+\frac{50\!\cdots\!09}{28\!\cdots\!68}a^{16}+\frac{24\!\cdots\!99}{14\!\cdots\!84}a^{14}+\frac{14\!\cdots\!75}{14\!\cdots\!84}a^{12}+\frac{19\!\cdots\!69}{56\!\cdots\!36}a^{10}+\frac{10\!\cdots\!53}{14\!\cdots\!84}a^{8}+\frac{23\!\cdots\!89}{28\!\cdots\!68}a^{6}+\frac{58\!\cdots\!31}{14\!\cdots\!84}a^{4}+\frac{21\!\cdots\!65}{70\!\cdots\!92}a^{2}+\frac{406372176616957}{35\!\cdots\!96}$, $\frac{38\!\cdots\!63}{13\!\cdots\!24}a^{23}-\frac{37\!\cdots\!95}{13\!\cdots\!24}a^{22}+\frac{30\!\cdots\!39}{13\!\cdots\!24}a^{21}-\frac{29\!\cdots\!43}{13\!\cdots\!24}a^{20}+\frac{23\!\cdots\!55}{33\!\cdots\!56}a^{19}-\frac{56\!\cdots\!55}{84\!\cdots\!64}a^{18}+\frac{76\!\cdots\!99}{67\!\cdots\!12}a^{17}-\frac{73\!\cdots\!99}{67\!\cdots\!12}a^{16}+\frac{72\!\cdots\!83}{67\!\cdots\!12}a^{15}-\frac{69\!\cdots\!71}{67\!\cdots\!12}a^{14}+\frac{20\!\cdots\!33}{33\!\cdots\!56}a^{13}-\frac{20\!\cdots\!35}{33\!\cdots\!56}a^{12}+\frac{29\!\cdots\!63}{13\!\cdots\!24}a^{11}-\frac{28\!\cdots\!39}{13\!\cdots\!24}a^{10}+\frac{59\!\cdots\!55}{13\!\cdots\!24}a^{9}-\frac{60\!\cdots\!95}{13\!\cdots\!24}a^{8}+\frac{16\!\cdots\!25}{33\!\cdots\!56}a^{7}-\frac{84\!\cdots\!83}{16\!\cdots\!28}a^{6}+\frac{41\!\cdots\!43}{21\!\cdots\!16}a^{5}-\frac{39\!\cdots\!65}{16\!\cdots\!28}a^{4}-\frac{28\!\cdots\!57}{21\!\cdots\!16}a^{3}-\frac{39\!\cdots\!63}{84\!\cdots\!64}a^{2}-\frac{25\!\cdots\!91}{84\!\cdots\!64}a-\frac{17\!\cdots\!91}{84\!\cdots\!64}$, $\frac{16\!\cdots\!95}{67\!\cdots\!12}a^{22}+\frac{12\!\cdots\!39}{67\!\cdots\!12}a^{20}+\frac{12\!\cdots\!77}{21\!\cdots\!16}a^{18}+\frac{31\!\cdots\!07}{33\!\cdots\!56}a^{16}+\frac{29\!\cdots\!07}{33\!\cdots\!56}a^{14}+\frac{81\!\cdots\!15}{16\!\cdots\!28}a^{12}+\frac{11\!\cdots\!59}{67\!\cdots\!12}a^{10}+\frac{21\!\cdots\!31}{67\!\cdots\!12}a^{8}+\frac{27\!\cdots\!19}{84\!\cdots\!64}a^{6}+\frac{31\!\cdots\!53}{22\!\cdots\!72}a^{4}+\frac{49\!\cdots\!79}{42\!\cdots\!32}a^{2}+\frac{74\!\cdots\!91}{42\!\cdots\!32}$ 423404988277887/854056124802492928a^23 + 32993890086790079/854056124802492928a^21 + 127825632590454015/106757015600311616a^19 + 8318885693004540311/427028062401246464a^17 + 79008367506209055511/427028062401246464a^15 + 228273325961919439691/213514031200623232a^13 + 3214760398677966881127/854056124802492928a^11 + 6654214590775656642427/854056124802492928a^9 + 20089841494307569929/2320804686963296a^7 + 430796747283810313973/106757015600311616a^5 + 6462741293582569005/53378507800155808a^3 - 737769868323733189/53378507800155808a, 41821863832804815/33735216929698470656a^23 - 3940387/22381668224a^22 + 815417920784696515/8433804232424617664a^21 - 153630661/11190834112a^20 + 101217184250894048319/33735216929698470656a^19 - 9532921237/22381668224a^18 + 412461708161938004171/8433804232424617664a^17 - 19417655015/2797708528a^16 + 3928083085664973901305/8433804232424617664a^15 - 184856597939/2797708528a^14 + 45586719899880274268417/16867608464849235328a^13 - 4288945569495/11190834112a^12 + 323177905056315662388759/33735216929698470656a^11 - 30397069709563/22381668224a^10 + 5289376028217414844105/263556382263269302a^9 - 31849594502383/11190834112a^8 + 33531914599259289191709/1466748562160803072a^7 - 72728437018777/22381668224a^6 + 193319713810621511942299/16867608464849235328a^5 - 18402326664749/11190834112a^4 + 8222560656854636788549/8433804232424617664a^3 - 847601185687/5595417056a^2 + 24814105207669558961/4216902116212308832a + 320629461/2797708528, 19654750614585/452821703754341888a^22 + 1532756204344557/452821703754341888a^20 + 185761494720595/1768834780290398a^18 + 387376045123214969/226410851877170944a^16 + 3685094622791720445/226410851877170944a^14 + 10667837547849315133/113205425938585472a^12 + 150586045800793525681/452821703754341888a^10 + 312595996243985745393/452821703754341888a^8 + 43572672567726400119/56602712969292736a^6 + 20379660628341634859/56602712969292736a^4 + 228631978754899561/28301356484646368a^2 - 3872984993458183/28301356484646368, 8240782615134225/16867608464849235328a^23 + 13538546485901/452821703754341888a^22 + 643114149790759861/16867608464849235328a^21 + 1061077929277279/452821703754341888a^20 + 39953897086218840697/33735216929698470656a^19 + 8289373900699711/113205425938585472a^18 + 326103876655384419301/16867608464849235328a^17 + 272807306249454645/226410851877170944a^16 + 1556199355710485449347/8433804232424617664a^15 + 2630750967412812091/226410851877170944a^14 + 18119076401465106132807/16867608464849235328a^13 + 7760225676824759109/113205425938585472a^12 + 64551064174515036558417/16867608464849235328a^11 + 112573362317561792037/452821703754341888a^10 + 136358590110405381077857/16867608464849235328a^9 + 244004724965122670563/452821703754341888a^8 + 13712578919874900876791/1466748562160803072a^7 + 73644895709684553791/113205425938585472a^6 + 81899245914170094794065/16867608464849235328a^5 + 657584127415868869/1768834780290398a^4 + 4320582836054166581379/8433804232424617664a^3 + 218517440820369673/3537669560580796a^2 - 13525343816674383609/4216902116212308832a + 3871631253850245/28301356484646368, 29631912857210403/8433804232424617664a^23 + 1326670075645485/8433804232424617664a^22 + 2311313362138341793/8433804232424617664a^21 + 12983662714322793/1054225529053077208a^20 + 71740732382273405037/8433804232424617664a^19 + 3241440150144127591/8433804232424617664a^18 + 1169747320334361362561/8433804232424617664a^17 + 26626131079447842901/4216902116212308832a^16 + 5572613677690680823279/4216902116212308832a^15 + 128141251271924225001/2108451058106154416a^14 + 32358234757509666032617/4216902116212308832a^13 + 1508363307999718473873/4216902116212308832a^12 + 229643840594851414713673/8433804232424617664a^11 + 10896553839331678781697/8433804232424617664a^10 + 481966059785741871953441/8433804232424617664a^9 + 5850107533068222797583/2108451058106154416a^8 + 550949854612492778162045/8433804232424617664a^7 + 27523234010429531527627/8433804232424617664a^6 + 278761818749691957649039/8433804232424617664a^5 + 450265386692813616883/263556382263269302a^4 + 6400303585983786489985/2108451058106154416a^3 + 339719702658227188623/2108451058106154416a^2 + 109124406946613594357/2108451058106154416a + 1541651057426990499/527112764526538604, 988953749313705249/134940867718793882624a^23 + 54252651155242895/134940867718793882624a^22 + 77137960468543624141/134940867718793882624a^21 + 4233735683613896363/134940867718793882624a^20 + 598555250762964094127/33735216929698470656a^19 + 8218239349424261283/8433804232424617664a^18 + 19518372357776060349573/67470433859396941312a^17 + 1072803304484688953551/67470433859396941312a^16 + 185957886063953496200605/67470433859396941312a^15 + 10229879838185776647739/67470433859396941312a^14 + 23471628845702915092417/1466748562160803072a^13 + 29724243632988325726211/33735216929698470656a^12 + 333106892215230846481743/5866994248643212288a^11 + 422174223389727635184791/134940867718793882624a^10 + 16076308716968568233070577/134940867718793882624a^9 + 886311041919891815793095/134940867718793882624a^8 + 4593181170303309547395629/33735216929698470656a^7 + 126598783852225840491999/16867608464849235328a^6 + 580953926462678191257717/8433804232424617664a^5 + 63918665465061396174013/16867608464849235328a^4 + 26779038215330359984907/4216902116212308832a^3 + 2885288665761567352379/8433804232424617664a^2 + 914221086786526390083/8433804232424617664a + 51706109763017627663/8433804232424617664, 280675339297154037/33735216929698470656a^23 + 21896042320322676639/33735216929698470656a^21 + 339887216134083183147/16867608464849235328a^19 + 5543647889623114837923/16867608464849235328a^17 + 52842941906303273693123/16867608464849235328a^15 + 38380511068502353219113/2108451058106154416a^13 + 2181325526006184613525773/33735216929698470656a^11 + 4586181696565216438783587/33735216929698470656a^9 + 114373179284148379751541/733374281080401536a^7 + 672005423830218934424265/8433804232424617664a^5 + 33302385862826078286895/4216902116212308832a^3 + 23126867044778375817/131778191131634651a, 133489120792725561/33735216929698470656a^23 - 3894544988883115/16867608464849235328a^22 + 5205679257124671521/16867608464849235328a^21 - 301994179372724263/16867608464849235328a^20 + 323115272379230749387/33735216929698470656a^19 - 12627415613307907/22917946283762548a^18 + 329214217118559292903/2108451058106154416a^17 - 74860822449387971595/8433804232424617664a^16 + 1567901745411325916533/1054225529053077208a^15 - 701637721624192652183/8433804232424617664a^14 + 145604008459116139739329/16867608464849235328a^13 - 1992483231253093151383/4216902116212308832a^12 + 1032629809131569242757825/33735216929698470656a^11 - 27456631289680801518739/16867608464849235328a^10 + 1082408071625958856438671/16867608464849235328a^9 - 55446640297223614536499/16867608464849235328a^8 + 2469947875411287055072103/33735216929698470656a^7 - 7568237227140539519609/2108451058106154416a^6 + 622554842447926056993547/16867608464849235328a^5 - 3687912693864056580129/2108451058106154416a^4 + 28134202541983620369809/8433804232424617664a^3 - 8948832865911139541/45835892567525096a^2 + 244919057951187702309/4216902116212308832a - 3481124339179949695/1054225529053077208, 25664562160333/56602712969292736a^22 + 250388646982603/7075339121161592a^20 + 31111664676666111/28301356484646368a^18 + 507725702551745409/28301356484646368a^16 + 2420671011756313399/14150678242323184a^14 + 14061823388467345875/14150678242323184a^12 + 199501780430669569669/56602712969292736a^10 + 104428584773712763553/14150678242323184a^8 + 237065707283690889989/28301356484646368a^6 + 58606570789320879431/14150678242323184a^4 + 2169195672568266565/7075339121161592a^2 + 406372176616957/3537669560580796, 38798024191602663/134940867718793882624a^23 - 37238609196430195/134940867718793882624a^22 + 3024331850485779339/134940867718793882624a^21 - 2901790727219961943/134940867718793882624a^20 + 23440876779384220055/33735216929698470656a^19 - 5622972960752219755/8433804232424617664a^18 + 762812531712557973499/67470433859396941312a^17 - 732697804941288731699/67470433859396941312a^16 + 7240379234333141434683/67470433859396941312a^15 - 6976165631318360954071/67470433859396941312a^14 + 20875324266614822280933/33735216929698470656a^13 - 20247423372985679527135/33735216929698470656a^12 + 292417313202131662586863/134940867718793882624a^11 - 287187195033752497904539/134940867718793882624a^10 + 597436567146603284181255/134940867718793882624a^9 - 600222522052536833996995/134940867718793882624a^8 + 160337455598714686812425/33735216929698470656a^7 - 84172601583254954466383/16867608464849235328a^6 + 4123335383562886354643/2108451058106154416a^5 - 39216689768105216240765/16867608464849235328a^4 - 288816273200942749757/2108451058106154416a^3 - 399311224698027516263/8433804232424617664a^2 - 25263465356152963391/8433804232424617664a - 1767890049569460691/8433804232424617664, 16627050649457095/67470433859396941312a^22 + 1285559928330693839/67470433859396941312a^20 + 1230958127385808677/2108451058106154416a^18 + 315247074068041679807/33735216929698470656a^16 + 2926572534060619295607/33735216929698470656a^14 + 8190220545425507903815/16867608464849235328a^12 + 110231481799061298601359/67470433859396941312a^10 + 213758100306780047954731/67470433859396941312a^8 + 27101675037579370836719/8433804232424617664a^6 + 310117772336972784553/227940654930395072a^4 + 493206240198568256179/4216902116212308832*a^2 + 7451597887361753291/4216902116212308832 (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:	$ 43654858009.901855 $ (assuming GRH)	sage: K.regulator() gp: K.reg magma: Regulator(K); oscar: regulator(K)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{12}\cdot 43654858009.901855 \cdot 1920}{6\cdot\sqrt{1704017867935867218637163902534560973508837376}}\cr\approx \mathstrut & 1.28116158011097 \end{aligned}\] (assuming GRH)

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

x = polygen(QQ); K.<a> = NumberField(x^24 + 78*x^22 + 2421*x^20 + 39474*x^18 + 376092*x^16 + 2183742*x^14 + 7748525*x^12 + 16261674*x^10 + 18591153*x^8 + 9415392*x^6 + 875496*x^4 + 15840*x^2 + 16)

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^24 + 78*x^22 + 2421*x^20 + 39474*x^18 + 376092*x^16 + 2183742*x^14 + 7748525*x^12 + 16261674*x^10 + 18591153*x^8 + 9415392*x^6 + 875496*x^4 + 15840*x^2 + 16, 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^24 + 78*x^22 + 2421*x^20 + 39474*x^18 + 376092*x^16 + 2183742*x^14 + 7748525*x^12 + 16261674*x^10 + 18591153*x^8 + 9415392*x^6 + 875496*x^4 + 15840*x^2 + 16);

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^24 + 78*x^22 + 2421*x^20 + 39474*x^18 + 376092*x^16 + 2183742*x^14 + 7748525*x^12 + 16261674*x^10 + 18591153*x^8 + 9415392*x^6 + 875496*x^4 + 15840*x^2 + 16);

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

$C_2^2\times D_6$ (as 24T30):

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 48

The 24 conjugacy class representatives for $C_2^2\times D_6$

Character table for $C_2^2\times D_6$ is not computed

Intermediate fields

$\Q(\sqrt{21}) $, $\Q(\sqrt{-3}) $, $\Q(\sqrt{14}) $, $\Q(\sqrt{-2}) $, $\Q(\sqrt{-7}) $, $\Q(\sqrt{6}) $, $\Q(\sqrt{-42}) $, 3.3.621.1, $\Q(\sqrt{6}, \sqrt{-7})$, $\Q(\sqrt{-3}, \sqrt{-7})$, $\Q(\sqrt{-2}, \sqrt{-7})$, $\Q(\sqrt{6}, \sqrt{14})$, $\Q(\sqrt{-2}, \sqrt{-3})$, $\Q(\sqrt{-3}, \sqrt{14})$, $\Q(\sqrt{-2}, \sqrt{21})$, 6.6.67724729856.1, 6.0.197448192.7, 6.6.396824589.1, 6.0.1156923.1, 6.0.132274863.5, 6.6.592344576.1, 6.0.203174189568.4, 8.0.796594176.1, 12.0.41279751306613600026624.4, 12.0.4586639034068177780736.1, 12.0.157469754435018921.1, 12.12.41279751306613600026624.2, 12.0.350872096716619776.1, 12.0.41279751306613600026624.3, 12.0.41279751306613600026624.6

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]

Sibling fields

Degree 24 siblings:	data not computed
Minimal sibling:	This field is its own minimal sibling

Frobenius cycle types

$p$	$2$	$3$	$5$	$7$	$11$	$13$	$17$	$19$	$23$	$29$	$31$	$37$	$41$	$43$	$47$	$53$	$59$
Cycle type	R	R	${\href{/padicField/5.6.0.1}{6} }^{4}$	R	${\href{/padicField/11.6.0.1}{6} }^{4}$	${\href{/padicField/13.6.0.1}{6} }^{4}$	${\href{/padicField/17.6.0.1}{6} }^{4}$	${\href{/padicField/19.2.0.1}{2} }^{12}$	R	${\href{/padicField/29.2.0.1}{2} }^{12}$	${\href{/padicField/31.6.0.1}{6} }^{4}$	${\href{/padicField/37.2.0.1}{2} }^{12}$	${\href{/padicField/41.2.0.1}{2} }^{12}$	${\href{/padicField/43.2.0.1}{2} }^{8}{,}\,{\href{/padicField/43.1.0.1}{1} }^{8}$	${\href{/padicField/47.2.0.1}{2} }^{12}$	${\href{/padicField/53.6.0.1}{6} }^{4}$	${\href{/padicField/59.2.0.1}{2} }^{12}$

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	2.4.6.2	$x^{4} + 4 x^{3} + 16 x^{2} + 24 x + 12$	$2$	$2$	$6$	$C_2^2$	$[3]^{2}$
	2.4.6.2	$x^{4} + 4 x^{3} + 16 x^{2} + 24 x + 12$	$2$	$2$	$6$	$C_2^2$	$[3]^{2}$
	2.4.6.2	$x^{4} + 4 x^{3} + 16 x^{2} + 24 x + 12$	$2$	$2$	$6$	$C_2^2$	$[3]^{2}$
	2.4.6.2	$x^{4} + 4 x^{3} + 16 x^{2} + 24 x + 12$	$2$	$2$	$6$	$C_2^2$	$[3]^{2}$
	2.4.6.2	$x^{4} + 4 x^{3} + 16 x^{2} + 24 x + 12$	$2$	$2$	$6$	$C_2^2$	$[3]^{2}$
	2.4.6.2	$x^{4} + 4 x^{3} + 16 x^{2} + 24 x + 12$	$2$	$2$	$6$	$C_2^2$	$[3]^{2}$
$3$ 3	3.12.14.6	$x^{12} + 6 x^{8} + 15 x^{6} + 9 x^{4} + 18 x^{2} + 9$	$6$	$2$	$14$	$D_6$	$[3/2]_{2}^{2}$
$3$ 3	3.12.14.6	$x^{12} + 6 x^{8} + 15 x^{6} + 9 x^{4} + 18 x^{2} + 9$	$6$	$2$	$14$	$D_6$	$[3/2]_{2}^{2}$
$7$ 7	7.4.2.1	$x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$
	7.4.2.1	$x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$
	7.4.2.1	$x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$
	7.4.2.1	$x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$
	7.4.2.1	$x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$
	7.4.2.1	$x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$
$23$ 23	23.2.0.1	$x^{2} + 21 x + 5$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
	23.2.0.1	$x^{2} + 21 x + 5$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
	23.2.0.1	$x^{2} + 21 x + 5$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
	23.2.0.1	$x^{2} + 21 x + 5$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
	23.4.2.1	$x^{4} + 42 x^{3} + 497 x^{2} + 1176 x + 10467$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$
	23.4.2.1	$x^{4} + 42 x^{3} + 497 x^{2} + 1176 x + 10467$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$
	23.4.2.1	$x^{4} + 42 x^{3} + 497 x^{2} + 1176 x + 10467$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$
	23.4.2.1	$x^{4} + 42 x^{3} + 497 x^{2} + 1176 x + 10467$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$

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