Properties

Label 24.4.153...449.2
Degree $24$
Signature $[4, 10]$
Discriminant $1.538\times 10^{33}$
Root discriminant $24.14$
Ramified prime $2083$
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 + 5*x^22 - 9*x^21 - 17*x^20 - 76*x^19 - 136*x^18 - 174*x^17 - 166*x^16 + 28*x^15 + 141*x^14 + 361*x^13 + 448*x^12 + 1384*x^11 + 3389*x^10 + 8270*x^9 + 11234*x^8 + 16185*x^7 + 17718*x^6 + 15868*x^5 + 15966*x^4 + 10226*x^3 + 4276*x^2 + 3258*x + 1459)
 
gp: K = bnfinit(x^24 + 5*x^22 - 9*x^21 - 17*x^20 - 76*x^19 - 136*x^18 - 174*x^17 - 166*x^16 + 28*x^15 + 141*x^14 + 361*x^13 + 448*x^12 + 1384*x^11 + 3389*x^10 + 8270*x^9 + 11234*x^8 + 16185*x^7 + 17718*x^6 + 15868*x^5 + 15966*x^4 + 10226*x^3 + 4276*x^2 + 3258*x + 1459, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1459, 3258, 4276, 10226, 15966, 15868, 17718, 16185, 11234, 8270, 3389, 1384, 448, 361, 141, 28, -166, -174, -136, -76, -17, -9, 5, 0, 1]);
 

\( x^{24} + 5 x^{22} - 9 x^{21} - 17 x^{20} - 76 x^{19} - 136 x^{18} - 174 x^{17} - 166 x^{16} + 28 x^{15} + 141 x^{14} + 361 x^{13} + 448 x^{12} + 1384 x^{11} + 3389 x^{10} + 8270 x^{9} + 11234 x^{8} + 16185 x^{7} + 17718 x^{6} + 15868 x^{5} + 15966 x^{4} + 10226 x^{3} + 4276 x^{2} + 3258 x + 1459 \)

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:  \(1537774657557415544813485393913449\)\(\medspace = 2083^{10}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $24.14$
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:  $2083$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $\frac{1}{73} a^{22} - \frac{11}{73} a^{21} + \frac{25}{73} a^{20} + \frac{24}{73} a^{19} - \frac{32}{73} a^{18} - \frac{31}{73} a^{17} + \frac{6}{73} a^{16} - \frac{29}{73} a^{15} - \frac{15}{73} a^{14} - \frac{17}{73} a^{13} + \frac{18}{73} a^{12} - \frac{18}{73} a^{11} - \frac{4}{73} a^{10} + \frac{34}{73} a^{9} - \frac{12}{73} a^{8} + \frac{4}{73} a^{7} - \frac{8}{73} a^{6} + \frac{28}{73} a^{5} - \frac{32}{73} a^{4} + \frac{33}{73} a^{3} + \frac{1}{73} a^{2} + \frac{20}{73} a + \frac{13}{73}$, $\frac{1}{1645186974275700917726662733962364567693322221381} a^{23} - \frac{2190995143971307031912721536334452333504228282}{1645186974275700917726662733962364567693322221381} a^{22} - \frac{437611615088791815081869014073698451444397387729}{1645186974275700917726662733962364567693322221381} a^{21} + \frac{647470821072303827519914056315736071560750362312}{1645186974275700917726662733962364567693322221381} a^{20} - \frac{568972904432962661656781422970404190656212434749}{1645186974275700917726662733962364567693322221381} a^{19} - \frac{634361912006356678772606004480768173003566233726}{1645186974275700917726662733962364567693322221381} a^{18} - \frac{735307982007087212042064917726163276377474067313}{1645186974275700917726662733962364567693322221381} a^{17} - \frac{46022256377659009554716072450941748150244470893}{1645186974275700917726662733962364567693322221381} a^{16} - \frac{662408069752058194263377944076141383319447392559}{1645186974275700917726662733962364567693322221381} a^{15} - \frac{433415563315763911589381092580682408318802936484}{1645186974275700917726662733962364567693322221381} a^{14} - \frac{808566614526914284850364694385424938649215725344}{1645186974275700917726662733962364567693322221381} a^{13} + \frac{383640571002589539601101490085166139028195570468}{1645186974275700917726662733962364567693322221381} a^{12} + \frac{113864821996663168146652519106170966340327742031}{1645186974275700917726662733962364567693322221381} a^{11} + \frac{300230490898256854529090000922011179684354331285}{1645186974275700917726662733962364567693322221381} a^{10} - \frac{26888886871913822954260320966397522808355999049}{1645186974275700917726662733962364567693322221381} a^{9} - \frac{688876708694360466139543795174717234698257904242}{1645186974275700917726662733962364567693322221381} a^{8} - \frac{555021248940417350798969262397516156631630635757}{1645186974275700917726662733962364567693322221381} a^{7} + \frac{574944885914486045175882517204428367823089355390}{1645186974275700917726662733962364567693322221381} a^{6} - \frac{29075605895655818606005775146043402420936983175}{1645186974275700917726662733962364567693322221381} a^{5} + \frac{796890991607973954782947333008153123554537699827}{1645186974275700917726662733962364567693322221381} a^{4} - \frac{123491659539365913760422851874766408966829512201}{1645186974275700917726662733962364567693322221381} a^{3} + \frac{59088243060726062631099153020395149541108643671}{1645186974275700917726662733962364567693322221381} a^{2} + \frac{585187777031209399765429072552081140870062601853}{1645186974275700917726662733962364567693322221381} a + \frac{702450411082488607747960685769666129201538310090}{1645186974275700917726662733962364567693322221381}$

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:  \( 10781013.84232622 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{4}\cdot(2\pi)^{10}\cdot 10781013.84232622 \cdot 1}{2\sqrt{1537774657557415544813485393913449}}\approx 0.210912308154547$ (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

6.2.4338889.1, 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: 24.4.1537774657557415544813485393913449.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type $20{,}\,{\href{/LocalNumberField/2.4.0.1}{4} }$ $20{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }$ $20{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }$ $20{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }$ ${\href{/LocalNumberField/11.12.0.1}{12} }^{2}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{4}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{6}$ $20{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{8}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{6}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{6}$ ${\href{/LocalNumberField/43.5.0.1}{5} }^{4}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{6}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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
2083Data not computed

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
1.2083.2t1.a.a$1$ $ 2083 $ \(\Q(\sqrt{-2083}) \) $C_2$ (as 2T1) $1$ $-1$
2.2083.120.b.a$2$ $ 2083 $ 24.4.1537774657557415544813485393913449.2 $C_4.A_5$ (as 24T576) $0$ $0$
2.2083.120.b.b$2$ $ 2083 $ 24.4.1537774657557415544813485393913449.2 $C_4.A_5$ (as 24T576) $0$ $0$
2.2083.120.b.c$2$ $ 2083 $ 24.4.1537774657557415544813485393913449.2 $C_4.A_5$ (as 24T576) $0$ $0$
2.2083.120.b.d$2$ $ 2083 $ 24.4.1537774657557415544813485393913449.2 $C_4.A_5$ (as 24T576) $0$ $0$
3.4338889.12t33.a.a$3$ $ 2083^{2}$ 5.1.4338889.1 $A_5$ (as 5T4) $1$ $-1$
3.4338889.12t33.a.b$3$ $ 2083^{2}$ 5.1.4338889.1 $A_5$ (as 5T4) $1$ $-1$
* 3.2083.12t76.a.a$3$ $ 2083 $ 10.0.39214470002250643.1 $A_5\times C_2$ (as 10T11) $1$ $1$
* 3.2083.12t76.a.b$3$ $ 2083 $ 10.0.39214470002250643.1 $A_5\times C_2$ (as 10T11) $1$ $1$
4.4338889.10t11.a.a$4$ $ 2083^{2}$ 10.0.39214470002250643.1 $A_5\times C_2$ (as 10T11) $1$ $0$
4.4338889.5t4.a.a$4$ $ 2083^{2}$ 5.1.4338889.1 $A_5$ (as 5T4) $1$ $0$
4.4338889.40t188.b.a$4$ $ 2083^{2}$ 24.4.1537774657557415544813485393913449.2 $C_4.A_5$ (as 24T576) $0$ $0$
4.4338889.40t188.b.b$4$ $ 2083^{2}$ 24.4.1537774657557415544813485393913449.2 $C_4.A_5$ (as 24T576) $0$ $0$
5.9037905787.12t75.a.a$5$ $ 2083^{3}$ 10.0.39214470002250643.1 $A_5\times C_2$ (as 10T11) $1$ $-1$
* 5.4338889.6t12.a.a$5$ $ 2083^{2}$ 5.1.4338889.1 $A_5$ (as 5T4) $1$ $1$
* 6.9037905787.24t576.b.a$6$ $ 2083^{3}$ 24.4.1537774657557415544813485393913449.2 $C_4.A_5$ (as 24T576) $0$ $0$
* 6.9037905787.24t576.b.b$6$ $ 2083^{3}$ 24.4.1537774657557415544813485393913449.2 $C_4.A_5$ (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 *.