Properties

Label 24.4.197...064.2
Degree $24$
Signature $[4, 10]$
Discriminant $1.976\times 10^{34}$
Root discriminant $26.85$
Ramified primes $2, 887$
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 - 11*x^23 + 53*x^22 - 131*x^21 + 131*x^20 + 144*x^19 - 585*x^18 + 471*x^17 + 846*x^16 - 3226*x^15 + 6073*x^14 - 7616*x^13 + 4239*x^12 + 5146*x^11 - 14303*x^10 + 15630*x^9 - 9886*x^8 + 3829*x^7 - 1007*x^6 + 240*x^5 - 15*x^4 - 39*x^3 + 19*x^2 - 3*x + 1)
 
gp: K = bnfinit(x^24 - 11*x^23 + 53*x^22 - 131*x^21 + 131*x^20 + 144*x^19 - 585*x^18 + 471*x^17 + 846*x^16 - 3226*x^15 + 6073*x^14 - 7616*x^13 + 4239*x^12 + 5146*x^11 - 14303*x^10 + 15630*x^9 - 9886*x^8 + 3829*x^7 - 1007*x^6 + 240*x^5 - 15*x^4 - 39*x^3 + 19*x^2 - 3*x + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -3, 19, -39, -15, 240, -1007, 3829, -9886, 15630, -14303, 5146, 4239, -7616, 6073, -3226, 846, 471, -585, 144, 131, -131, 53, -11, 1]);
 

\( x^{24} - 11 x^{23} + 53 x^{22} - 131 x^{21} + 131 x^{20} + 144 x^{19} - 585 x^{18} + 471 x^{17} + 846 x^{16} - 3226 x^{15} + 6073 x^{14} - 7616 x^{13} + 4239 x^{12} + 5146 x^{11} - 14303 x^{10} + 15630 x^{9} - 9886 x^{8} + 3829 x^{7} - 1007 x^{6} + 240 x^{5} - 15 x^{4} - 39 x^{3} + 19 x^{2} - 3 x + 1 \)

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:  \(19756778413055716819205133752664064\)\(\medspace = 2^{16}\cdot 887^{10}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $26.85$
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:  $2, 887$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{2} a^{18} - \frac{1}{2} a^{16} - \frac{1}{2} a^{15} - \frac{1}{2} a^{12} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{19} - \frac{1}{2} a^{17} - \frac{1}{2} a^{16} - \frac{1}{2} a^{13} - \frac{1}{2} a^{12} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{20} - \frac{1}{2} a^{17} - \frac{1}{2} a^{16} - \frac{1}{2} a^{15} - \frac{1}{2} a^{14} - \frac{1}{2} a^{13} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{21} - \frac{1}{2} a^{17} - \frac{1}{2} a^{14} - \frac{1}{2} a^{12} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{22} - \frac{1}{2} a^{16} - \frac{1}{2} a^{13} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{734875002403804819351514058255122} a^{23} + \frac{52527180720531312782089213442467}{367437501201902409675757029127561} a^{22} - \frac{128858027033476485067405523109275}{734875002403804819351514058255122} a^{21} + \frac{51933286546121821592948209032869}{734875002403804819351514058255122} a^{20} + \frac{53408547151287390963031269081425}{367437501201902409675757029127561} a^{19} - \frac{76455412351971200021086398659473}{367437501201902409675757029127561} a^{18} - \frac{12550891723131108778390767582313}{734875002403804819351514058255122} a^{17} - \frac{172766575632896549374670639343327}{734875002403804819351514058255122} a^{16} + \frac{81715108034567303699652284244241}{734875002403804819351514058255122} a^{15} + \frac{148584816296962143661857253771603}{734875002403804819351514058255122} a^{14} - \frac{300113323466706870981422181899705}{734875002403804819351514058255122} a^{13} + \frac{237508640984049361645207776862291}{734875002403804819351514058255122} a^{12} + \frac{97755250964274970117835334585471}{734875002403804819351514058255122} a^{11} - \frac{101161976728521577006751080390701}{367437501201902409675757029127561} a^{10} - \frac{13544998300887236871843652243664}{367437501201902409675757029127561} a^{9} + \frac{60147841296112989148164441931749}{734875002403804819351514058255122} a^{8} - \frac{110066360940465980619837380374237}{734875002403804819351514058255122} a^{7} + \frac{56996181849411659469861744961901}{367437501201902409675757029127561} a^{6} + \frac{180855740564491382237066909130551}{734875002403804819351514058255122} a^{5} - \frac{57453034900841302362907770325379}{367437501201902409675757029127561} a^{4} - \frac{97256006948571726161369547965294}{367437501201902409675757029127561} a^{3} + \frac{30190398514476596717978349955624}{367437501201902409675757029127561} a^{2} + \frac{66861399103974620059379785711468}{367437501201902409675757029127561} a - \frac{42506773650306783785854255813850}{367437501201902409675757029127561}$

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 $2$)
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:  \( 33313220.242250312 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{4}\cdot(2\pi)^{10}\cdot 33313220.242250312 \cdot 1}{2\sqrt{19756778413055716819205133752664064}}\approx 0.181822331396691$ (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

6.2.12588304.1, 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: 24.4.19756778413055716819205133752664064.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{2}$ $20{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{8}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{6}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{8}$ $20{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{6}$ ${\href{/LocalNumberField/23.12.0.1}{12} }^{2}$ ${\href{/LocalNumberField/29.12.0.1}{12} }^{2}$ ${\href{/LocalNumberField/31.12.0.1}{12} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{4}$ $20{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{6}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{4}$ $20{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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
$2$2.12.8.1$x^{12} - 6 x^{9} + 12 x^{6} - 8 x^{3} + 16$$3$$4$$8$$C_3 : C_4$$[\ ]_{3}^{4}$
2.12.8.1$x^{12} - 6 x^{9} + 12 x^{6} - 8 x^{3} + 16$$3$$4$$8$$C_3 : C_4$$[\ ]_{3}^{4}$
887Data not computed

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
1.887.2t1.a.a$1$ $ 887 $ \(\Q(\sqrt{-887}) \) $C_2$ (as 2T1) $1$ $-1$
2.3548.120.b.a$2$ $ 2^{2} \cdot 887 $ 24.4.19756778413055716819205133752664064.2 $C_4.A_5$ (as 24T576) $0$ $0$
2.3548.120.b.b$2$ $ 2^{2} \cdot 887 $ 24.4.19756778413055716819205133752664064.2 $C_4.A_5$ (as 24T576) $0$ $0$
2.3548.120.b.c$2$ $ 2^{2} \cdot 887 $ 24.4.19756778413055716819205133752664064.2 $C_4.A_5$ (as 24T576) $0$ $0$
2.3548.120.b.d$2$ $ 2^{2} \cdot 887 $ 24.4.19756778413055716819205133752664064.2 $C_4.A_5$ (as 24T576) $0$ $0$
3.3147076.12t33.a.a$3$ $ 2^{2} \cdot 887^{2}$ 5.1.3147076.1 $A_5$ (as 5T4) $1$ $-1$
3.3147076.12t33.a.b$3$ $ 2^{2} \cdot 887^{2}$ 5.1.3147076.1 $A_5$ (as 5T4) $1$ $-1$
* 3.3548.12t76.a.a$3$ $ 2^{2} \cdot 887 $ 10.0.8784925479251312.1 $A_5\times C_2$ (as 10T11) $1$ $1$
* 3.3548.12t76.a.b$3$ $ 2^{2} \cdot 887 $ 10.0.8784925479251312.1 $A_5\times C_2$ (as 10T11) $1$ $1$
4.3147076.10t11.a.a$4$ $ 2^{2} \cdot 887^{2}$ 10.0.8784925479251312.1 $A_5\times C_2$ (as 10T11) $1$ $0$
4.3147076.5t4.a.a$4$ $ 2^{2} \cdot 887^{2}$ 5.1.3147076.1 $A_5$ (as 5T4) $1$ $0$
4.3147076.40t188.b.a$4$ $ 2^{2} \cdot 887^{2}$ 24.4.19756778413055716819205133752664064.2 $C_4.A_5$ (as 24T576) $0$ $0$
4.3147076.40t188.b.b$4$ $ 2^{2} \cdot 887^{2}$ 24.4.19756778413055716819205133752664064.2 $C_4.A_5$ (as 24T576) $0$ $0$
5.11165825648.12t75.a.a$5$ $ 2^{4} \cdot 887^{3}$ 10.0.8784925479251312.1 $A_5\times C_2$ (as 10T11) $1$ $-1$
* 5.12588304.6t12.a.a$5$ $ 2^{4} \cdot 887^{2}$ 5.1.3147076.1 $A_5$ (as 5T4) $1$ $1$
* 6.11165825648.24t576.b.a$6$ $ 2^{4} \cdot 887^{3}$ 24.4.19756778413055716819205133752664064.2 $C_4.A_5$ (as 24T576) $0$ $0$
* 6.11165825648.24t576.b.b$6$ $ 2^{4} \cdot 887^{3}$ 24.4.19756778413055716819205133752664064.2 $C_4.A_5$ (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 *.