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Properties

Label 24.4.153...449.1
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 - 2*x^23 - 8*x^22 - 2*x^21 + 70*x^20 - 33*x^19 - 156*x^18 + 43*x^17 + 433*x^16 - 998*x^15 + 433*x^14 + 1850*x^13 - 1996*x^12 - 1334*x^11 + 1462*x^10 + 5*x^9 - 1195*x^8 - 1071*x^7 - 747*x^6 - 160*x^5 + 26*x^4 - 66*x^3 - 57*x^2 - 14*x - 1)
 
gp: K = bnfinit(x^24 - 2*x^23 - 8*x^22 - 2*x^21 + 70*x^20 - 33*x^19 - 156*x^18 + 43*x^17 + 433*x^16 - 998*x^15 + 433*x^14 + 1850*x^13 - 1996*x^12 - 1334*x^11 + 1462*x^10 + 5*x^9 - 1195*x^8 - 1071*x^7 - 747*x^6 - 160*x^5 + 26*x^4 - 66*x^3 - 57*x^2 - 14*x - 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, -14, -57, -66, 26, -160, -747, -1071, -1195, 5, 1462, -1334, -1996, 1850, 433, -998, 433, 43, -156, -33, 70, -2, -8, -2, 1]);
 

\( x^{24} - 2 x^{23} - 8 x^{22} - 2 x^{21} + 70 x^{20} - 33 x^{19} - 156 x^{18} + 43 x^{17} + 433 x^{16} - 998 x^{15} + 433 x^{14} + 1850 x^{13} - 1996 x^{12} - 1334 x^{11} + 1462 x^{10} + 5 x^{9} - 1195 x^{8} - 1071 x^{7} - 747 x^{6} - 160 x^{5} + 26 x^{4} - 66 x^{3} - 57 x^{2} - 14 x - 1 \)

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:  \(1537774657557415544813485393913449\)\(\medspace = 2083^{10}\)
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $\frac{1}{84753373039324165201925412594029144171} a^{23} + \frac{8826023875837578237495068875450125018}{84753373039324165201925412594029144171} a^{22} - \frac{18195891377689533246038707133039710192}{84753373039324165201925412594029144171} a^{21} - \frac{36784127225931776510871373427696546569}{84753373039324165201925412594029144171} a^{20} - \frac{1160848353530740843194809581596939587}{84753373039324165201925412594029144171} a^{19} + \frac{25720156941615916758599673842900234061}{84753373039324165201925412594029144171} a^{18} + \frac{34695964771168874717309517082133579216}{84753373039324165201925412594029144171} a^{17} + \frac{14838771522168440654475303473990151841}{84753373039324165201925412594029144171} a^{16} - \frac{37333446656838321848722494305516475623}{84753373039324165201925412594029144171} a^{15} + \frac{18801675178973428378571128028046385934}{84753373039324165201925412594029144171} a^{14} + \frac{15327781705989055493730315918875991483}{84753373039324165201925412594029144171} a^{13} + \frac{201095899419391379281334040252559862}{1971008675333120120975009595209980097} a^{12} + \frac{20127971428182742773030339471667864363}{84753373039324165201925412594029144171} a^{11} - \frac{12228390566164592011860362152138181610}{84753373039324165201925412594029144171} a^{10} + \frac{31135257756950843355068856748511838415}{84753373039324165201925412594029144171} a^{9} + \frac{20803630907021541493938781994448825790}{84753373039324165201925412594029144171} a^{8} - \frac{25043423249496814082198785418350069427}{84753373039324165201925412594029144171} a^{7} + \frac{36138825608250072343597162565325053966}{84753373039324165201925412594029144171} a^{6} + \frac{1293260086785472224486435996836969425}{84753373039324165201925412594029144171} a^{5} - \frac{38132379068565402472593533146179307427}{84753373039324165201925412594029144171} a^{4} + \frac{38628755150631384487290990776692832971}{84753373039324165201925412594029144171} a^{3} - \frac{37834450791030664206845852580851419461}{84753373039324165201925412594029144171} a^{2} - \frac{32661795568825269556617284138442061884}{84753373039324165201925412594029144171} a - \frac{33301543378776042200603292264127584748}{84753373039324165201925412594029144171}$

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: 24.4.1537774657557415544813485393913449.2

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.a.a$2$ $ 2083 $ 24.4.1537774657557415544813485393913449.1 $C_4.A_5$ (as 24T576) $0$ $0$
2.2083.120.a.b$2$ $ 2083 $ 24.4.1537774657557415544813485393913449.1 $C_4.A_5$ (as 24T576) $0$ $0$
2.2083.120.a.c$2$ $ 2083 $ 24.4.1537774657557415544813485393913449.1 $C_4.A_5$ (as 24T576) $0$ $0$
2.2083.120.a.d$2$ $ 2083 $ 24.4.1537774657557415544813485393913449.1 $C_4.A_5$ (as 24T576) $0$ $0$
* 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) $1$ $1$
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) $1$ $-1$
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.a.a$4$ $ 2083^{2}$ 24.4.1537774657557415544813485393913449.1 $C_4.A_5$ (as 24T576) $0$ $0$
4.4338889.40t188.a.b$4$ $ 2083^{2}$ 24.4.1537774657557415544813485393913449.1 $C_4.A_5$ (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.a.a$6$ $ 2083^{3}$ 24.4.1537774657557415544813485393913449.1 $C_4.A_5$ (as 24T576) $0$ $0$
* 6.9037905787.24t576.a.b$6$ $ 2083^{3}$ 24.4.1537774657557415544813485393913449.1 $C_4.A_5$ (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 *.