LMFDB - Number field 24.0.23583143319155607774002556174336000000000000.1

Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^24 + 2*x^22 - 19*x^20 - 92*x^18 + 68*x^16 + 1152*x^14 + 1920*x^12 + 18432*x^10 + 17408*x^8 - 376832*x^6 - 1245184*x^4 + 2097152*x^2 + 16777216)

gp: K = bnfinit(y^24 + 2*y^22 - 19*y^20 - 92*y^18 + 68*y^16 + 1152*y^14 + 1920*y^12 + 18432*y^10 + 17408*y^8 - 376832*y^6 - 1245184*y^4 + 2097152*y^2 + 16777216, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^24 + 2*x^22 - 19*x^20 - 92*x^18 + 68*x^16 + 1152*x^14 + 1920*x^12 + 18432*x^10 + 17408*x^8 - 376832*x^6 - 1245184*x^4 + 2097152*x^2 + 16777216);

oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^24 + 2*x^22 - 19*x^20 - 92*x^18 + 68*x^16 + 1152*x^14 + 1920*x^12 + 18432*x^10 + 17408*x^8 - 376832*x^6 - 1245184*x^4 + 2097152*x^2 + 16777216)

$ x^{24} + 2 x^{22} - 19 x^{20} - 92 x^{18} + 68 x^{16} + 1152 x^{14} + 1920 x^{12} + 18432 x^{10} + \cdots + 16777216 $ x^24 + 2*x^22 - 19*x^20 - 92*x^18 + 68*x^16 + 1152*x^14 + 1920*x^12 + 18432*x^10 + 17408*x^8 - 376832*x^6 - 1245184*x^4 + 2097152*x^2 + 16777216

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:	$23583143319155607774002556174336000000000000$ 23583143319155607774002556174336000000000000 $\medspace = 2^{48}\cdot 3^{12}\cdot 5^{12}\cdot 71^{8}$	sage: K.disc() gp: K.disc magma: OK := Integers(K); Discriminant(OK); oscar: OK = ring_of_integers(K); discriminant(OK)
Root discriminant:	$64.15$	sage: (K.disc().abs())^(1./K.degree()) gp: abs(K.disc)^(1/poldegree(K.pol)) magma: Abs(Discriminant(OK))^(1/Degree(K)); oscar: (1.0 * dK)^(1/degree(K))
Galois root discriminant:	$2^{2}3^{1/2}5^{1/2}71^{1/2}\approx 130.53735097664577$
Ramified primes:	$2$, $3$, $5$, $71$ 2, 3, 5, 71	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 $a$)

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2}a^{4}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{5}-\frac{1}{2}a^{3}$, $\frac{1}{4}a^{6}-\frac{1}{4}a^{5}-\frac{1}{4}a^{4}-\frac{1}{4}a^{3}-\frac{1}{2}a$, $\frac{1}{4}a^{7}+\frac{1}{4}a^{3}-\frac{1}{2}a$, $\frac{1}{8}a^{8}+\frac{1}{8}a^{4}-\frac{1}{4}a^{2}$, $\frac{1}{16}a^{9}-\frac{3}{16}a^{5}+\frac{1}{8}a^{3}-\frac{1}{2}a$, $\frac{1}{64}a^{10}+\frac{5}{64}a^{6}-\frac{1}{4}a^{5}+\frac{5}{32}a^{4}+\frac{1}{4}a^{3}-\frac{1}{8}a^{2}$, $\frac{1}{64}a^{11}+\frac{5}{64}a^{7}-\frac{3}{32}a^{5}-\frac{3}{8}a^{3}-\frac{1}{2}a$, $\frac{1}{128}a^{12}+\frac{5}{128}a^{8}-\frac{3}{64}a^{6}-\frac{1}{4}a^{5}-\frac{3}{16}a^{4}+\frac{1}{4}a^{3}+\frac{1}{4}a^{2}$, $\frac{1}{256}a^{13}-\frac{1}{128}a^{11}+\frac{5}{256}a^{9}-\frac{1}{16}a^{7}-\frac{11}{64}a^{5}+\frac{3}{16}a^{3}-\frac{1}{2}a$, $\frac{1}{1024}a^{14}-\frac{1}{512}a^{13}-\frac{1}{512}a^{12}+\frac{1}{256}a^{11}+\frac{5}{1024}a^{10}+\frac{11}{512}a^{9}+\frac{3}{64}a^{8}-\frac{3}{32}a^{7}+\frac{21}{256}a^{6}+\frac{31}{128}a^{5}+\frac{15}{64}a^{4}+\frac{3}{32}a^{3}-\frac{1}{2}a^{2}+\frac{1}{4}a$, $\frac{1}{2048}a^{15}+\frac{1}{1024}a^{13}-\frac{3}{2048}a^{11}+\frac{1}{512}a^{9}+\frac{5}{512}a^{7}-\frac{1}{4}a^{5}-\frac{3}{32}a^{3}+\frac{1}{4}a$, $\frac{1}{16384}a^{16}+\frac{1}{8192}a^{14}+\frac{29}{16384}a^{12}-\frac{31}{4096}a^{10}-\frac{211}{4096}a^{8}-\frac{5}{256}a^{6}-\frac{1}{4}a^{5}+\frac{61}{256}a^{4}-\frac{1}{4}a^{3}+\frac{9}{32}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{65536}a^{17}+\frac{1}{32768}a^{15}-\frac{99}{65536}a^{13}+\frac{33}{16384}a^{11}+\frac{141}{16384}a^{9}-\frac{37}{1024}a^{7}+\frac{245}{1024}a^{5}+\frac{21}{128}a^{3}+\frac{3}{8}a$, $\frac{1}{262144}a^{18}+\frac{1}{131072}a^{16}-\frac{99}{262144}a^{14}-\frac{223}{65536}a^{12}+\frac{141}{65536}a^{10}-\frac{1}{32}a^{9}+\frac{139}{4096}a^{8}+\frac{341}{4096}a^{6}-\frac{5}{32}a^{5}+\frac{101}{512}a^{4}-\frac{5}{16}a^{3}+\frac{11}{32}a^{2}+\frac{1}{4}a$, $\frac{1}{524288}a^{19}+\frac{1}{262144}a^{17}-\frac{99}{524288}a^{15}-\frac{223}{131072}a^{13}+\frac{141}{131072}a^{11}+\frac{139}{8192}a^{9}+\frac{341}{8192}a^{7}-\frac{155}{1024}a^{5}-\frac{5}{64}a^{3}$, $\frac{1}{35651584}a^{20}-\frac{1}{1048576}a^{19}+\frac{21}{17825792}a^{18}+\frac{3}{524288}a^{17}+\frac{557}{35651584}a^{16}-\frac{141}{1048576}a^{15}+\frac{2659}{8912896}a^{14}-\frac{103}{262144}a^{13}-\frac{34299}{8912896}a^{12}-\frac{1733}{262144}a^{11}-\frac{3105}{1114112}a^{10}+\frac{855}{32768}a^{9}-\frac{16415}{557056}a^{8}-\frac{1357}{16384}a^{7}-\frac{3461}{34816}a^{6}-\frac{29}{128}a^{5}+\frac{67}{2176}a^{4}-\frac{21}{256}a^{3}+\frac{543}{1088}a^{2}-\frac{7}{16}a-\frac{15}{68}$, $\frac{1}{71303168}a^{21}-\frac{13}{35651584}a^{19}-\frac{123}{71303168}a^{17}-\frac{1}{32768}a^{16}-\frac{141}{8912896}a^{15}-\frac{1}{16384}a^{14}+\frac{20441}{17825792}a^{13}-\frac{29}{32768}a^{12}+\frac{7577}{4456448}a^{11}+\frac{31}{8192}a^{10}-\frac{20869}{1114112}a^{9}+\frac{211}{8192}a^{8}-\frac{13793}{278528}a^{7}-\frac{59}{512}a^{6}+\frac{1543}{17408}a^{5}+\frac{3}{512}a^{4}+\frac{1239}{4352}a^{3}-\frac{9}{64}a^{2}-\frac{47}{272}a-\frac{1}{4}$, $\frac{1}{285212672}a^{22}-\frac{1}{142606336}a^{20}-\frac{1}{1048576}a^{19}-\frac{203}{285212672}a^{18}+\frac{3}{524288}a^{17}+\frac{5}{1048576}a^{16}-\frac{141}{1048576}a^{15}+\frac{2385}{71303168}a^{14}-\frac{103}{262144}a^{13}+\frac{38151}{17825792}a^{12}-\frac{1733}{262144}a^{11}-\frac{13861}{4456448}a^{10}+\frac{855}{32768}a^{9}+\frac{52685}{1114112}a^{8}-\frac{1357}{16384}a^{7}+\frac{1021}{17408}a^{6}+\frac{3}{128}a^{5}+\frac{2643}{17408}a^{4}-\frac{85}{256}a^{3}-\frac{305}{1088}a^{2}-\frac{7}{16}a+\frac{3}{34}$, $\frac{1}{570425344}a^{23}-\frac{1}{285212672}a^{21}+\frac{341}{570425344}a^{19}-\frac{7}{2097152}a^{17}-\frac{1}{32768}a^{16}+\frac{21561}{142606336}a^{15}-\frac{1}{16384}a^{14}-\frac{17473}{35651584}a^{13}-\frac{29}{32768}a^{12}+\frac{10245}{8912896}a^{11}+\frac{31}{8192}a^{10}+\frac{42417}{2228224}a^{9}+\frac{211}{8192}a^{8}+\frac{29061}{278528}a^{7}-\frac{59}{512}a^{6}+\frac{3187}{34816}a^{5}+\frac{3}{512}a^{4}-\frac{661}{4352}a^{3}-\frac{9}{64}a^{2}+\frac{63}{272}a-\frac{1}{4}$ 1, a, a^2, a^3, 1/2*a^4 - 1/2*a^2, 1/2*a^5 - 1/2*a^3, 1/4*a^6 - 1/4*a^5 - 1/4*a^4 - 1/4*a^3 - 1/2*a, 1/4*a^7 + 1/4*a^3 - 1/2*a, 1/8*a^8 + 1/8*a^4 - 1/4*a^2, 1/16*a^9 - 3/16*a^5 + 1/8*a^3 - 1/2*a, 1/64*a^10 + 5/64*a^6 - 1/4*a^5 + 5/32*a^4 + 1/4*a^3 - 1/8*a^2, 1/64*a^11 + 5/64*a^7 - 3/32*a^5 - 3/8*a^3 - 1/2*a, 1/128*a^12 + 5/128*a^8 - 3/64*a^6 - 1/4*a^5 - 3/16*a^4 + 1/4*a^3 + 1/4*a^2, 1/256*a^13 - 1/128*a^11 + 5/256*a^9 - 1/16*a^7 - 11/64*a^5 + 3/16*a^3 - 1/2*a, 1/1024*a^14 - 1/512*a^13 - 1/512*a^12 + 1/256*a^11 + 5/1024*a^10 + 11/512*a^9 + 3/64*a^8 - 3/32*a^7 + 21/256*a^6 + 31/128*a^5 + 15/64*a^4 + 3/32*a^3 - 1/2*a^2 + 1/4*a, 1/2048*a^15 + 1/1024*a^13 - 3/2048*a^11 + 1/512*a^9 + 5/512*a^7 - 1/4*a^5 - 3/32*a^3 + 1/4*a, 1/16384*a^16 + 1/8192*a^14 + 29/16384*a^12 - 31/4096*a^10 - 211/4096*a^8 - 5/256*a^6 - 1/4*a^5 + 61/256*a^4 - 1/4*a^3 + 9/32*a^2 - 1/2*a - 1/2, 1/65536*a^17 + 1/32768*a^15 - 99/65536*a^13 + 33/16384*a^11 + 141/16384*a^9 - 37/1024*a^7 + 245/1024*a^5 + 21/128*a^3 + 3/8*a, 1/262144*a^18 + 1/131072*a^16 - 99/262144*a^14 - 223/65536*a^12 + 141/65536*a^10 - 1/32*a^9 + 139/4096*a^8 + 341/4096*a^6 - 5/32*a^5 + 101/512*a^4 - 5/16*a^3 + 11/32*a^2 + 1/4*a, 1/524288*a^19 + 1/262144*a^17 - 99/524288*a^15 - 223/131072*a^13 + 141/131072*a^11 + 139/8192*a^9 + 341/8192*a^7 - 155/1024*a^5 - 5/64*a^3, 1/35651584*a^20 - 1/1048576*a^19 + 21/17825792*a^18 + 3/524288*a^17 + 557/35651584*a^16 - 141/1048576*a^15 + 2659/8912896*a^14 - 103/262144*a^13 - 34299/8912896*a^12 - 1733/262144*a^11 - 3105/1114112*a^10 + 855/32768*a^9 - 16415/557056*a^8 - 1357/16384*a^7 - 3461/34816*a^6 - 29/128*a^5 + 67/2176*a^4 - 21/256*a^3 + 543/1088*a^2 - 7/16*a - 15/68, 1/71303168*a^21 - 13/35651584*a^19 - 123/71303168*a^17 - 1/32768*a^16 - 141/8912896*a^15 - 1/16384*a^14 + 20441/17825792*a^13 - 29/32768*a^12 + 7577/4456448*a^11 + 31/8192*a^10 - 20869/1114112*a^9 + 211/8192*a^8 - 13793/278528*a^7 - 59/512*a^6 + 1543/17408*a^5 + 3/512*a^4 + 1239/4352*a^3 - 9/64*a^2 - 47/272*a - 1/4, 1/285212672*a^22 - 1/142606336*a^20 - 1/1048576*a^19 - 203/285212672*a^18 + 3/524288*a^17 + 5/1048576*a^16 - 141/1048576*a^15 + 2385/71303168*a^14 - 103/262144*a^13 + 38151/17825792*a^12 - 1733/262144*a^11 - 13861/4456448*a^10 + 855/32768*a^9 + 52685/1114112*a^8 - 1357/16384*a^7 + 1021/17408*a^6 + 3/128*a^5 + 2643/17408*a^4 - 85/256*a^3 - 305/1088*a^2 - 7/16*a + 3/34, 1/570425344*a^23 - 1/285212672*a^21 + 341/570425344*a^19 - 7/2097152*a^17 - 1/32768*a^16 + 21561/142606336*a^15 - 1/16384*a^14 - 17473/35651584*a^13 - 29/32768*a^12 + 10245/8912896*a^11 + 31/8192*a^10 + 42417/2228224*a^9 + 211/8192*a^8 + 29061/278528*a^7 - 59/512*a^6 + 3187/34816*a^5 + 3/512*a^4 - 661/4352*a^3 - 9/64*a^2 + 63/272*a - 1/4

sage: K.integral_basis()

gp: K.zk

magma: IntegralBasis(K);

oscar: basis(OK)

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

Class group and class number

$C_{4}\times C_{4}\times C_{88}$, which has order $1408$ (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:	$ -1 $ -1 (order $2$)	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:	$\frac{99}{285212672}a^{23}-\frac{13}{71303168}a^{22}+\frac{31}{142606336}a^{21}+\frac{13}{35651584}a^{20}-\frac{2377}{285212672}a^{19}+\frac{463}{71303168}a^{18}-\frac{175}{71303168}a^{17}+\frac{7}{262144}a^{16}+\frac{2995}{71303168}a^{15}+\frac{3}{17825792}a^{14}+\frac{291}{8912896}a^{13}-\frac{515}{4456448}a^{12}+\frac{803}{1114112}a^{11}-\frac{687}{1114112}a^{10}+\frac{3599}{557056}a^{9}-\frac{1675}{278528}a^{8}-\frac{863}{278528}a^{7}-\frac{13}{4352}a^{6}-\frac{1111}{8704}a^{5}+\frac{1033}{8704}a^{4}+\frac{657}{4352}a^{3}+\frac{407}{1088}a^{2}+\frac{171}{272}a+\frac{11}{68}$, $\frac{11}{17825792}a^{22}+\frac{67}{8912896}a^{20}+\frac{647}{17825792}a^{18}-\frac{581}{4456448}a^{16}-\frac{3919}{4456448}a^{14}-\frac{29}{32768}a^{12}-\frac{97}{557056}a^{10}+\frac{1519}{69632}a^{8}+\frac{6089}{34816}a^{6}+\frac{21}{32}a^{4}-\frac{63}{32}a^{2}-\frac{367}{34}$, $\frac{245}{285212672}a^{23}+\frac{69}{142606336}a^{22}-\frac{27}{142606336}a^{21}-\frac{205}{71303168}a^{20}-\frac{7215}{285212672}a^{19}-\frac{951}{142606336}a^{18}+\frac{2165}{71303168}a^{17}+\frac{3}{32768}a^{16}+\frac{18421}{71303168}a^{15}-\frac{4755}{35651584}a^{14}+\frac{1}{34816}a^{13}+\frac{1159}{8912896}a^{12}+\frac{1245}{2228224}a^{11}-\frac{1519}{2228224}a^{10}+\frac{2859}{278528}a^{9}+\frac{3385}{557056}a^{8}-\frac{6543}{278528}a^{7}-\frac{3783}{69632}a^{6}-\frac{3641}{8704}a^{5}-\frac{519}{8704}a^{4}+\frac{3669}{4352}a^{3}+\frac{1685}{1088}a^{2}+\frac{927}{272}a-\frac{175}{68}$, $\frac{35}{71303168}a^{23}-\frac{61}{35651584}a^{22}+\frac{353}{71303168}a^{21}-\frac{149}{17825792}a^{20}+\frac{801}{71303168}a^{19}-\frac{225}{35651584}a^{18}-\frac{2525}{71303168}a^{17}+\frac{1355}{8912896}a^{16}-\frac{8101}{17825792}a^{15}+\frac{5289}{8912896}a^{14}-\frac{10389}{17825792}a^{13}+\frac{47}{65536}a^{12}-\frac{3839}{4456448}a^{11}+\frac{303}{1114112}a^{10}+\frac{13101}{1114112}a^{9}-\frac{5617}{139264}a^{8}+\frac{27735}{278528}a^{7}-\frac{11351}{69632}a^{6}+\frac{4919}{17408}a^{5}-\frac{1}{4}a^{4}-\frac{1357}{4352}a^{3}+\frac{113}{64}a^{2}-\frac{1255}{272}a+\frac{227}{68}$, $\frac{1}{4194304}a^{23}-\frac{1}{2097152}a^{22}+\frac{3}{2097152}a^{21}-\frac{3}{1048576}a^{20}+\frac{5}{4194304}a^{19}-\frac{5}{2097152}a^{18}-\frac{9}{524288}a^{17}+\frac{9}{262144}a^{16}-\frac{55}{1048576}a^{15}+\frac{55}{524288}a^{14}+\frac{17}{262144}a^{13}-\frac{17}{131072}a^{12}+\frac{47}{65536}a^{11}-\frac{47}{32768}a^{10}+\frac{119}{16384}a^{9}-\frac{119}{8192}a^{8}+\frac{17}{512}a^{7}-\frac{17}{256}a^{6}+\frac{11}{256}a^{5}-\frac{11}{128}a^{4}-\frac{1}{8}a^{3}+\frac{1}{4}a^{2}+1$, $\frac{25}{71303168}a^{23}-\frac{67}{35651584}a^{22}+\frac{325}{71303168}a^{21}+\frac{37}{17825792}a^{20}-\frac{1089}{71303168}a^{19}+\frac{2785}{35651584}a^{18}-\frac{6413}{71303168}a^{17}-\frac{195}{8912896}a^{16}+\frac{3413}{17825792}a^{15}-\frac{9017}{8912896}a^{14}+\frac{8195}{17825792}a^{13}-\frac{51}{65536}a^{12}-\frac{2611}{4456448}a^{11}+\frac{121}{1114112}a^{10}+\frac{8809}{1114112}a^{9}-\frac{3639}{139264}a^{8}+\frac{17115}{278528}a^{7}+\frac{7743}{69632}a^{6}-\frac{2407}{8704}a^{5}+\frac{73}{64}a^{4}-\frac{3563}{4352}a^{3}-\frac{89}{64}a^{2}+\frac{891}{272}a-\frac{803}{68}$, $\frac{25}{71303168}a^{23}+\frac{67}{35651584}a^{22}+\frac{325}{71303168}a^{21}-\frac{37}{17825792}a^{20}-\frac{1089}{71303168}a^{19}-\frac{2785}{35651584}a^{18}-\frac{6413}{71303168}a^{17}+\frac{195}{8912896}a^{16}+\frac{3413}{17825792}a^{15}+\frac{9017}{8912896}a^{14}+\frac{8195}{17825792}a^{13}+\frac{51}{65536}a^{12}-\frac{2611}{4456448}a^{11}-\frac{121}{1114112}a^{10}+\frac{8809}{1114112}a^{9}+\frac{3639}{139264}a^{8}+\frac{17115}{278528}a^{7}-\frac{7743}{69632}a^{6}-\frac{2407}{8704}a^{5}-\frac{73}{64}a^{4}-\frac{3563}{4352}a^{3}+\frac{89}{64}a^{2}+\frac{891}{272}a+\frac{803}{68}$, $\frac{155}{285212672}a^{23}+\frac{53}{142606336}a^{21}-\frac{4745}{285212672}a^{19}-\frac{19}{17825792}a^{17}+\frac{6611}{71303168}a^{15}+\frac{2349}{17825792}a^{13}+\frac{6659}{4456448}a^{11}+\frac{8535}{1114112}a^{9}-\frac{1571}{139264}a^{7}-\frac{1307}{4352}a^{5}+\frac{409}{1088}a^{3}+\frac{29}{17}a$, $\frac{63}{142606336}a^{23}-\frac{67}{35651584}a^{22}-\frac{63}{35651584}a^{21}+\frac{37}{17825792}a^{20}+\frac{1639}{142606336}a^{19}+\frac{2785}{35651584}a^{18}+\frac{1901}{71303168}a^{17}-\frac{195}{8912896}a^{16}-\frac{10801}{35651584}a^{15}-\frac{9017}{8912896}a^{14}-\frac{3293}{17825792}a^{13}-\frac{51}{65536}a^{12}+\frac{3197}{4456448}a^{11}+\frac{121}{1114112}a^{10}+\frac{1613}{1114112}a^{9}-\frac{3639}{139264}a^{8}-\frac{4151}{278528}a^{7}+\frac{7743}{69632}a^{6}+\frac{5323}{17408}a^{5}+\frac{73}{64}a^{4}+\frac{1429}{4352}a^{3}-\frac{89}{64}a^{2}-\frac{1461}{272}a-\frac{871}{68}$, $\frac{645}{142606336}a^{23}-\frac{369}{142606336}a^{22}+\frac{1}{4456448}a^{21}+\frac{641}{71303168}a^{20}-\frac{18427}{142606336}a^{19}+\frac{4731}{142606336}a^{18}-\frac{655}{71303168}a^{17}-\frac{55}{524288}a^{16}+\frac{36993}{35651584}a^{15}-\frac{10073}{35651584}a^{14}+\frac{22131}{17825792}a^{13}-\frac{3895}{8912896}a^{12}+\frac{19827}{4456448}a^{11}+\frac{599}{2228224}a^{10}+\frac{73337}{1114112}a^{9}-\frac{17721}{557056}a^{8}-\frac{21661}{278528}a^{7}+\frac{11209}{69632}a^{6}-\frac{37459}{17408}a^{5}+\frac{5463}{8704}a^{4}+\frac{5827}{4352}a^{3}-\frac{2931}{1088}a^{2}+\frac{4789}{272}a-\frac{195}{68}$, $\frac{263}{285212672}a^{23}-\frac{379}{71303168}a^{22}+\frac{1123}{142606336}a^{21}-\frac{1171}{35651584}a^{20}+\frac{4555}{285212672}a^{19}-\frac{1311}{71303168}a^{18}-\frac{5883}{71303168}a^{17}+\frac{8897}{17825792}a^{16}-\frac{40889}{71303168}a^{15}+\frac{37197}{17825792}a^{14}-\frac{13245}{8912896}a^{13}+\frac{213}{65536}a^{12}-\frac{5}{34816}a^{11}+\frac{39}{69632}a^{10}+\frac{16059}{557056}a^{9}-\frac{8095}{69632}a^{8}+\frac{46273}{278528}a^{7}-\frac{48113}{69632}a^{6}+\frac{2097}{8704}a^{5}-\frac{99}{128}a^{4}-\frac{4703}{4352}a^{3}+\frac{315}{64}a^{2}-\frac{929}{272}a+\frac{1081}{68}$ 99/285212672a^23 - 13/71303168a^22 + 31/142606336a^21 + 13/35651584a^20 - 2377/285212672a^19 + 463/71303168a^18 - 175/71303168a^17 + 7/262144a^16 + 2995/71303168a^15 + 3/17825792a^14 + 291/8912896a^13 - 515/4456448a^12 + 803/1114112a^11 - 687/1114112a^10 + 3599/557056a^9 - 1675/278528a^8 - 863/278528a^7 - 13/4352a^6 - 1111/8704a^5 + 1033/8704a^4 + 657/4352a^3 + 407/1088a^2 + 171/272a + 11/68, 11/17825792a^22 + 67/8912896a^20 + 647/17825792a^18 - 581/4456448a^16 - 3919/4456448a^14 - 29/32768a^12 - 97/557056a^10 + 1519/69632a^8 + 6089/34816a^6 + 21/32a^4 - 63/32a^2 - 367/34, 245/285212672a^23 + 69/142606336a^22 - 27/142606336a^21 - 205/71303168a^20 - 7215/285212672a^19 - 951/142606336a^18 + 2165/71303168a^17 + 3/32768a^16 + 18421/71303168a^15 - 4755/35651584a^14 + 1/34816a^13 + 1159/8912896a^12 + 1245/2228224a^11 - 1519/2228224a^10 + 2859/278528a^9 + 3385/557056a^8 - 6543/278528a^7 - 3783/69632a^6 - 3641/8704a^5 - 519/8704a^4 + 3669/4352a^3 + 1685/1088a^2 + 927/272a - 175/68, 35/71303168a^23 - 61/35651584a^22 + 353/71303168a^21 - 149/17825792a^20 + 801/71303168a^19 - 225/35651584a^18 - 2525/71303168a^17 + 1355/8912896a^16 - 8101/17825792a^15 + 5289/8912896a^14 - 10389/17825792a^13 + 47/65536a^12 - 3839/4456448a^11 + 303/1114112a^10 + 13101/1114112a^9 - 5617/139264a^8 + 27735/278528a^7 - 11351/69632a^6 + 4919/17408a^5 - 1/4a^4 - 1357/4352a^3 + 113/64a^2 - 1255/272a + 227/68, 1/4194304a^23 - 1/2097152a^22 + 3/2097152a^21 - 3/1048576a^20 + 5/4194304a^19 - 5/2097152a^18 - 9/524288a^17 + 9/262144a^16 - 55/1048576a^15 + 55/524288a^14 + 17/262144a^13 - 17/131072a^12 + 47/65536a^11 - 47/32768a^10 + 119/16384a^9 - 119/8192a^8 + 17/512a^7 - 17/256a^6 + 11/256a^5 - 11/128a^4 - 1/8a^3 + 1/4a^2 + 1, 25/71303168a^23 - 67/35651584a^22 + 325/71303168a^21 + 37/17825792a^20 - 1089/71303168a^19 + 2785/35651584a^18 - 6413/71303168a^17 - 195/8912896a^16 + 3413/17825792a^15 - 9017/8912896a^14 + 8195/17825792a^13 - 51/65536a^12 - 2611/4456448a^11 + 121/1114112a^10 + 8809/1114112a^9 - 3639/139264a^8 + 17115/278528a^7 + 7743/69632a^6 - 2407/8704a^5 + 73/64a^4 - 3563/4352a^3 - 89/64a^2 + 891/272a - 803/68, 25/71303168a^23 + 67/35651584a^22 + 325/71303168a^21 - 37/17825792a^20 - 1089/71303168a^19 - 2785/35651584a^18 - 6413/71303168a^17 + 195/8912896a^16 + 3413/17825792a^15 + 9017/8912896a^14 + 8195/17825792a^13 + 51/65536a^12 - 2611/4456448a^11 - 121/1114112a^10 + 8809/1114112a^9 + 3639/139264a^8 + 17115/278528a^7 - 7743/69632a^6 - 2407/8704a^5 - 73/64a^4 - 3563/4352a^3 + 89/64a^2 + 891/272a + 803/68, 155/285212672a^23 + 53/142606336a^21 - 4745/285212672a^19 - 19/17825792a^17 + 6611/71303168a^15 + 2349/17825792a^13 + 6659/4456448a^11 + 8535/1114112a^9 - 1571/139264a^7 - 1307/4352a^5 + 409/1088a^3 + 29/17a, 63/142606336a^23 - 67/35651584a^22 - 63/35651584a^21 + 37/17825792a^20 + 1639/142606336a^19 + 2785/35651584a^18 + 1901/71303168a^17 - 195/8912896a^16 - 10801/35651584a^15 - 9017/8912896a^14 - 3293/17825792a^13 - 51/65536a^12 + 3197/4456448a^11 + 121/1114112a^10 + 1613/1114112a^9 - 3639/139264a^8 - 4151/278528a^7 + 7743/69632a^6 + 5323/17408a^5 + 73/64a^4 + 1429/4352a^3 - 89/64a^2 - 1461/272a - 871/68, 645/142606336a^23 - 369/142606336a^22 + 1/4456448a^21 + 641/71303168a^20 - 18427/142606336a^19 + 4731/142606336a^18 - 655/71303168a^17 - 55/524288a^16 + 36993/35651584a^15 - 10073/35651584a^14 + 22131/17825792a^13 - 3895/8912896a^12 + 19827/4456448a^11 + 599/2228224a^10 + 73337/1114112a^9 - 17721/557056a^8 - 21661/278528a^7 + 11209/69632a^6 - 37459/17408a^5 + 5463/8704a^4 + 5827/4352a^3 - 2931/1088a^2 + 4789/272a - 195/68, 263/285212672a^23 - 379/71303168a^22 + 1123/142606336a^21 - 1171/35651584a^20 + 4555/285212672a^19 - 1311/71303168a^18 - 5883/71303168a^17 + 8897/17825792a^16 - 40889/71303168a^15 + 37197/17825792a^14 - 13245/8912896a^13 + 213/65536a^12 - 5/34816a^11 + 39/69632a^10 + 16059/557056a^9 - 8095/69632a^8 + 46273/278528a^7 - 48113/69632a^6 + 2097/8704a^5 - 99/128a^4 - 4703/4352a^3 + 315/64a^2 - 929/272*a + 1081/68 (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:	$ 27910062769.535168 $ (assuming GRH)	sage: K.regulator() gp: K.reg magma: Regulator(K); oscar: regulator(K)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{12}\cdot 27910062769.535168 \cdot 1408}{2\cdot\sqrt{23583143319155607774002556174336000000000000}}\cr\approx \mathstrut & 15.3176114003772 \end{aligned}\] (assuming GRH)

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

x = polygen(QQ); K.<a> = NumberField(x^24 + 2*x^22 - 19*x^20 - 92*x^18 + 68*x^16 + 1152*x^14 + 1920*x^12 + 18432*x^10 + 17408*x^8 - 376832*x^6 - 1245184*x^4 + 2097152*x^2 + 16777216)

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 + 2*x^22 - 19*x^20 - 92*x^18 + 68*x^16 + 1152*x^14 + 1920*x^12 + 18432*x^10 + 17408*x^8 - 376832*x^6 - 1245184*x^4 + 2097152*x^2 + 16777216, 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 + 2*x^22 - 19*x^20 - 92*x^18 + 68*x^16 + 1152*x^14 + 1920*x^12 + 18432*x^10 + 17408*x^8 - 376832*x^6 - 1245184*x^4 + 2097152*x^2 + 16777216);

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 + 2*x^22 - 19*x^20 - 92*x^18 + 68*x^16 + 1152*x^14 + 1920*x^12 + 18432*x^10 + 17408*x^8 - 376832*x^6 - 1245184*x^4 + 2097152*x^2 + 16777216);

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 $C_2^2\times D_6$

Intermediate fields

$\Q(\sqrt{3}) $, $\Q(\sqrt{-15}) $, $\Q(\sqrt{10}) $, $\Q(\sqrt{-2}) $, $\Q(\sqrt{-5}) $, $\Q(\sqrt{30}) $, $\Q(\sqrt{-6}) $, 3.3.568.1, $\Q(\sqrt{-5}, \sqrt{-6})$, $\Q(\sqrt{3}, \sqrt{-5})$, $\Q(\sqrt{-2}, \sqrt{-5})$, $\Q(\sqrt{3}, \sqrt{10})$, $\Q(\sqrt{-2}, \sqrt{-15})$, $\Q(\sqrt{-2}, \sqrt{3})$, $\Q(\sqrt{-6}, \sqrt{10})$, 6.0.1088856000.6, 6.6.139373568.1, 6.0.2580992.1, 6.6.1290496000.1, 6.0.645248000.3, 6.6.8710848000.1, 6.0.278747136.2, 8.0.3317760000.1, 12.0.4856247864262656000000.1, 12.0.303515491516416000000.1, 12.0.6661519704064000000.1, 12.0.75878872879104000000.1, 12.12.4856247864262656000000.1, 12.0.1214061966065664000000.1, 12.0.310799863312809984.1

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	R	${\href{/padicField/7.6.0.1}{6} }^{4}$	${\href{/padicField/11.2.0.1}{2} }^{12}$	${\href{/padicField/13.6.0.1}{6} }^{4}$	${\href{/padicField/17.2.0.1}{2} }^{12}$	${\href{/padicField/19.6.0.1}{6} }^{4}$	${\href{/padicField/23.6.0.1}{6} }^{4}$	${\href{/padicField/29.2.0.1}{2} }^{12}$	${\href{/padicField/31.6.0.1}{6} }^{4}$	${\href{/padicField/37.2.0.1}{2} }^{12}$	${\href{/padicField/41.2.0.1}{2} }^{12}$	${\href{/padicField/43.6.0.1}{6} }^{4}$	${\href{/padicField/47.6.0.1}{6} }^{4}$	${\href{/padicField/53.2.0.1}{2} }^{12}$	${\href{/padicField/59.2.0.1}{2} }^{12}$

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	2.4.8.4	$x^{4} + 6 x^{2} + 4 x + 6$	$4$	$1$	$8$	$C_2^2$	$[2, 3]$
	2.4.8.4	$x^{4} + 6 x^{2} + 4 x + 6$	$4$	$1$	$8$	$C_2^2$	$[2, 3]$
	2.4.8.4	$x^{4} + 6 x^{2} + 4 x + 6$	$4$	$1$	$8$	$C_2^2$	$[2, 3]$
	2.4.8.4	$x^{4} + 6 x^{2} + 4 x + 6$	$4$	$1$	$8$	$C_2^2$	$[2, 3]$
	2.4.8.4	$x^{4} + 6 x^{2} + 4 x + 6$	$4$	$1$	$8$	$C_2^2$	$[2, 3]$
	2.4.8.4	$x^{4} + 6 x^{2} + 4 x + 6$	$4$	$1$	$8$	$C_2^2$	$[2, 3]$
$3$ 3	3.6.3.1	$x^{6} + 18 x^{2} - 27$	$2$	$3$	$3$	$C_6$	$[\ ]_{2}^{3}$
	3.6.3.1	$x^{6} + 18 x^{2} - 27$	$2$	$3$	$3$	$C_6$	$[\ ]_{2}^{3}$
	3.6.3.1	$x^{6} + 18 x^{2} - 27$	$2$	$3$	$3$	$C_6$	$[\ ]_{2}^{3}$
	3.6.3.1	$x^{6} + 18 x^{2} - 27$	$2$	$3$	$3$	$C_6$	$[\ ]_{2}^{3}$
$5$ 5	5.4.2.1	$x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$
	5.4.2.1	$x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$
	5.4.2.1	$x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$
	5.4.2.1	$x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$
	5.4.2.1	$x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$
	5.4.2.1	$x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$
$71$ 71	71.2.0.1	$x^{2} + 69 x + 7$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
	71.2.0.1	$x^{2} + 69 x + 7$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
	71.2.0.1	$x^{2} + 69 x + 7$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
	71.2.0.1	$x^{2} + 69 x + 7$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
	71.4.2.1	$x^{4} + 138 x^{3} + 4917 x^{2} + 10764 x + 342127$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$
	71.4.2.1	$x^{4} + 138 x^{3} + 4917 x^{2} + 10764 x + 342127$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$
	71.4.2.1	$x^{4} + 138 x^{3} + 4917 x^{2} + 10764 x + 342127$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$
	71.4.2.1	$x^{4} + 138 x^{3} + 4917 x^{2} + 10764 x + 342127$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$

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