Properties

Label 24.4.940...625.1
Degree $24$
Signature $[4, 10]$
Discriminant $9.404\times 10^{32}$
Root discriminant $23.65$
Ramified primes $5, 151$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $C_4.A_5$ (as 24T576)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^24 - 10*x^23 + 48*x^22 - 158*x^21 + 381*x^20 - 714*x^19 + 1213*x^18 - 1907*x^17 + 2614*x^16 - 3516*x^15 + 4694*x^14 - 5466*x^13 + 6009*x^12 - 6785*x^11 + 7506*x^10 - 8491*x^9 + 9365*x^8 - 8498*x^7 + 5899*x^6 - 3118*x^5 + 1203*x^4 - 296*x^3 + 36*x^2 - 7*x + 1)
 
gp: K = bnfinit(x^24 - 10*x^23 + 48*x^22 - 158*x^21 + 381*x^20 - 714*x^19 + 1213*x^18 - 1907*x^17 + 2614*x^16 - 3516*x^15 + 4694*x^14 - 5466*x^13 + 6009*x^12 - 6785*x^11 + 7506*x^10 - 8491*x^9 + 9365*x^8 - 8498*x^7 + 5899*x^6 - 3118*x^5 + 1203*x^4 - 296*x^3 + 36*x^2 - 7*x + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -7, 36, -296, 1203, -3118, 5899, -8498, 9365, -8491, 7506, -6785, 6009, -5466, 4694, -3516, 2614, -1907, 1213, -714, 381, -158, 48, -10, 1]);
 

\(x^{24} - 10 x^{23} + 48 x^{22} - 158 x^{21} + 381 x^{20} - 714 x^{19} + 1213 x^{18} - 1907 x^{17} + 2614 x^{16} - 3516 x^{15} + 4694 x^{14} - 5466 x^{13} + 6009 x^{12} - 6785 x^{11} + 7506 x^{10} - 8491 x^{9} + 9365 x^{8} - 8498 x^{7} + 5899 x^{6} - 3118 x^{5} + 1203 x^{4} - 296 x^{3} + 36 x^{2} - 7 x + 1\)  Toggle raw display

sage: K.defining_polynomial()
 
gp: K.pol
 
magma: DefiningPolynomial(K);
 

Invariants

Degree:  $24$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[4, 10]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(940350029043078386535797119140625\)\(\medspace = 5^{16}\cdot 151^{10}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $23.65$
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
Ramified primes:  $5, 151$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Aut(K/\Q)|$:  $4$
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $\frac{1}{875125528187897786812793362957451372} a^{23} + \frac{297910072207355670197174735889822049}{875125528187897786812793362957451372} a^{22} + \frac{148950382269433624985505638798611159}{875125528187897786812793362957451372} a^{21} - \frac{7882921600039810103706538010118705}{875125528187897786812793362957451372} a^{20} + \frac{218024731943997684834650575595184783}{437562764093948893406396681478725686} a^{19} + \frac{23167770797040626358634591102764461}{218781382046974446703198340739362843} a^{18} + \frac{28070856480447493611161740005146177}{875125528187897786812793362957451372} a^{17} + \frac{9923486054539175961157635486784438}{218781382046974446703198340739362843} a^{16} - \frac{208245976926690287853629926757934385}{437562764093948893406396681478725686} a^{15} - \frac{116653483207228527812261328145429615}{437562764093948893406396681478725686} a^{14} + \frac{66033036153168352389090994015335289}{218781382046974446703198340739362843} a^{13} + \frac{121383692978433991653456962546743617}{437562764093948893406396681478725686} a^{12} - \frac{391181227056777335934856971000790033}{875125528187897786812793362957451372} a^{11} + \frac{72632260741017513731530661855819739}{218781382046974446703198340739362843} a^{10} - \frac{43129641678651491115513993228108915}{437562764093948893406396681478725686} a^{9} - \frac{297969163008158790181692585847983657}{875125528187897786812793362957451372} a^{8} + \frac{54321822701238036573178679071410253}{437562764093948893406396681478725686} a^{7} - \frac{44163883788644414331338698569065754}{218781382046974446703198340739362843} a^{6} + \frac{197678238846539460674978159237830267}{875125528187897786812793362957451372} a^{5} - \frac{73613892314370036057193815451950893}{875125528187897786812793362957451372} a^{4} + \frac{94091296039126895924835987494231607}{218781382046974446703198340739362843} a^{3} - \frac{92462216976377579456038917821066977}{218781382046974446703198340739362843} a^{2} - \frac{106513713495078713465352275663587523}{218781382046974446703198340739362843} a - \frac{289574885819262321608123177625323211}{875125528187897786812793362957451372}$  Toggle raw display

sage: K.integral_basis()
 
gp: K.zk
 
magma: IntegralBasis(K);
 

Class group and class number

Trivial group, which has order $1$ (assuming GRH)

sage: K.class_group().invariants()
 
gp: K.clgp
 
magma: ClassGroup(K);
 

Unit group

sage: UK = K.unit_group()
 
magma: UK, f := UnitGroup(K);
 
Rank:  $13$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( -1 \) (order $2$)  Toggle raw display
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
Fundamental units:  Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH)
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 8548806.54922059 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{4}\cdot(2\pi)^{10}\cdot 8548806.54922059 \cdot 1}{2\sqrt{940350029043078386535797119140625}}\approx 0.213869763394454$ (assuming GRH)

Galois group

$C_4.A_5$ (as 24T576):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A non-solvable group of order 240
The 18 conjugacy class representatives for $C_4.A_5$
Character table for $C_4.A_5$

Intermediate fields

6.2.14250625.2, Deg 12

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 40 siblings: data not computed
Arithmetically equvalently sibling: 24.4.940350029043078386535797119140625.2

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/2.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/3.12.0.1}{12} }^{2}$ R $20{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{6}$ $20{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{8}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/23.12.0.1}{12} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{6}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ $20{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{8}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ $20{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{4}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
gp: idealfactors = idealprimedec(K, p); \\ get the data
 
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])
 
magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
magma: idealfactors := Factorization(p*Integers(K)); // get the data
 
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$5$5.12.8.1$x^{12} - 30 x^{9} + 175 x^{6} + 500 x^{3} + 5000$$3$$4$$8$$C_3 : C_4$$[\ ]_{3}^{4}$
5.12.8.1$x^{12} - 30 x^{9} + 175 x^{6} + 500 x^{3} + 5000$$3$$4$$8$$C_3 : C_4$$[\ ]_{3}^{4}$
$151$151.4.2.2$x^{4} - 151 x^{2} + 273612$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
151.4.0.1$x^{4} - x + 6$$1$$4$$0$$C_4$$[\ ]^{4}$
151.8.4.1$x^{8} + 273612 x^{4} - 3442951 x^{2} + 18715881636$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
151.8.4.1$x^{8} + 273612 x^{4} - 3442951 x^{2} + 18715881636$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
1.151.2t1.a.a$1$ $ 151 $ \(\Q(\sqrt{-151}) \) $C_2$ (as 2T1) $1$ $-1$
2.3775.120.a.a$2$ $ 5^{2} \cdot 151 $ 24.4.940350029043078386535797119140625.1 $C_4.A_5$ (as 24T576) $0$ $0$
2.3775.120.a.b$2$ $ 5^{2} \cdot 151 $ 24.4.940350029043078386535797119140625.1 $C_4.A_5$ (as 24T576) $0$ $0$
2.3775.120.a.c$2$ $ 5^{2} \cdot 151 $ 24.4.940350029043078386535797119140625.1 $C_4.A_5$ (as 24T576) $0$ $0$
2.3775.120.a.d$2$ $ 5^{2} \cdot 151 $ 24.4.940350029043078386535797119140625.1 $C_4.A_5$ (as 24T576) $0$ $0$
* 3.3775.12t76.a.a$3$ $ 5^{2} \cdot 151 $ 10.0.49064203594375.1 $A_5\times C_2$ (as 10T11) $1$ $1$
* 3.3775.12t76.a.b$3$ $ 5^{2} \cdot 151 $ 10.0.49064203594375.1 $A_5\times C_2$ (as 10T11) $1$ $1$
3.570025.12t33.a.a$3$ $ 5^{2} \cdot 151^{2}$ 5.1.570025.1 $A_5$ (as 5T4) $1$ $-1$
3.570025.12t33.a.b$3$ $ 5^{2} \cdot 151^{2}$ 5.1.570025.1 $A_5$ (as 5T4) $1$ $-1$
4.570025.10t11.a.a$4$ $ 5^{2} \cdot 151^{2}$ 10.0.49064203594375.1 $A_5\times C_2$ (as 10T11) $1$ $0$
4.570025.5t4.a.a$4$ $ 5^{2} \cdot 151^{2}$ 5.1.570025.1 $A_5$ (as 5T4) $1$ $0$
4.570025.40t188.a.a$4$ $ 5^{2} \cdot 151^{2}$ 24.4.940350029043078386535797119140625.1 $C_4.A_5$ (as 24T576) $0$ $0$
4.570025.40t188.a.b$4$ $ 5^{2} \cdot 151^{2}$ 24.4.940350029043078386535797119140625.1 $C_4.A_5$ (as 24T576) $0$ $0$
5.2151844375.12t75.a.a$5$ $ 5^{4} \cdot 151^{3}$ 10.0.49064203594375.1 $A_5\times C_2$ (as 10T11) $1$ $-1$
* 5.14250625.6t12.a.a$5$ $ 5^{4} \cdot 151^{2}$ 5.1.570025.1 $A_5$ (as 5T4) $1$ $1$
* 6.2151844375.24t576.a.a$6$ $ 5^{4} \cdot 151^{3}$ 24.4.940350029043078386535797119140625.1 $C_4.A_5$ (as 24T576) $0$ $0$
* 6.2151844375.24t576.a.b$6$ $ 5^{4} \cdot 151^{3}$ 24.4.940350029043078386535797119140625.1 $C_4.A_5$ (as 24T576) $0$ $0$

Data is given for all irreducible representations of the Galois group for the Galois closure of this field. Those marked with * are summands in the permutation representation coming from this field. Representations which appear with multiplicity greater than one are indicated by exponents on the *.