Properties

Label 24.4.940...625.2
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 - 2*x^23 + 6*x^22 - 5*x^21 + 11*x^20 - 17*x^19 + 2*x^18 - 34*x^17 - 139*x^16 - 108*x^15 - 455*x^14 - 497*x^13 - 1230*x^12 - 1691*x^11 - 2719*x^10 - 4214*x^9 - 4936*x^8 - 6130*x^7 - 5653*x^6 - 4317*x^5 - 2323*x^4 - 45*x^3 - 323*x^2 + 428*x + 173)
 
gp: K = bnfinit(x^24 - 2*x^23 + 6*x^22 - 5*x^21 + 11*x^20 - 17*x^19 + 2*x^18 - 34*x^17 - 139*x^16 - 108*x^15 - 455*x^14 - 497*x^13 - 1230*x^12 - 1691*x^11 - 2719*x^10 - 4214*x^9 - 4936*x^8 - 6130*x^7 - 5653*x^6 - 4317*x^5 - 2323*x^4 - 45*x^3 - 323*x^2 + 428*x + 173, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![173, 428, -323, -45, -2323, -4317, -5653, -6130, -4936, -4214, -2719, -1691, -1230, -497, -455, -108, -139, -34, 2, -17, 11, -5, 6, -2, 1]);
 

\( x^{24} - 2 x^{23} + 6 x^{22} - 5 x^{21} + 11 x^{20} - 17 x^{19} + 2 x^{18} - 34 x^{17} - 139 x^{16} - 108 x^{15} - 455 x^{14} - 497 x^{13} - 1230 x^{12} - 1691 x^{11} - 2719 x^{10} - 4214 x^{9} - 4936 x^{8} - 6130 x^{7} - 5653 x^{6} - 4317 x^{5} - 2323 x^{4} - 45 x^{3} - 323 x^{2} + 428 x + 173 \)

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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $\frac{1}{2} a^{22} - \frac{1}{2} a^{21} - \frac{1}{2} a^{18} - \frac{1}{2} a^{16} - \frac{1}{2} a^{15} - \frac{1}{2} a^{14} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{24498727281240124099191850974287478308578} a^{23} + \frac{3710546243226076921268272146468673232115}{24498727281240124099191850974287478308578} a^{22} + \frac{2162952239244527749060904971796472318542}{12249363640620062049595925487143739154289} a^{21} - \frac{4811109892166050828140277310153613736482}{12249363640620062049595925487143739154289} a^{20} - \frac{10765438924583734877531073850815090368363}{24498727281240124099191850974287478308578} a^{19} + \frac{3100580018102610156573838314651515062406}{12249363640620062049595925487143739154289} a^{18} - \frac{3343566034031005483447072348777349358105}{24498727281240124099191850974287478308578} a^{17} - \frac{8043399870115749266766434103093803967009}{24498727281240124099191850974287478308578} a^{16} - \frac{545734595493220870334534730799110855129}{24498727281240124099191850974287478308578} a^{15} - \frac{114955875602119953368765649520291582374}{12249363640620062049595925487143739154289} a^{14} - \frac{3966365215076830032697675217068564392620}{12249363640620062049595925487143739154289} a^{13} - \frac{9461552836209748896572059537978715047367}{24498727281240124099191850974287478308578} a^{12} - \frac{7866207705673987369742032820424940367879}{24498727281240124099191850974287478308578} a^{11} - \frac{2744275593818821877568097911070076973883}{24498727281240124099191850974287478308578} a^{10} - \frac{12047515744516666234192974341744274494129}{24498727281240124099191850974287478308578} a^{9} + \frac{2520541965945126198589918572883051746756}{12249363640620062049595925487143739154289} a^{8} + \frac{2079415946876799135598444416166717784647}{24498727281240124099191850974287478308578} a^{7} + \frac{344933479362339271858239468800174870643}{24498727281240124099191850974287478308578} a^{6} - \frac{7709844818670604822128212406971079337819}{24498727281240124099191850974287478308578} a^{5} - \frac{10809525086350149983066780640962498816827}{24498727281240124099191850974287478308578} a^{4} - \frac{6570155719911128348526113973607789609473}{24498727281240124099191850974287478308578} a^{3} - \frac{11227901057787607693251768658948612907973}{24498727281240124099191850974287478308578} a^{2} - \frac{11949101136443058328685344869177250653321}{24498727281240124099191850974287478308578} a + \frac{5931863792038136416045704260123183601826}{12249363640620062049595925487143739154289}$

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:  \( 8548806.54922059 \) (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: 24.4.940350029043078386535797119140625.1

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.b.a$2$ $ 5^{2} \cdot 151 $ 24.4.940350029043078386535797119140625.2 $C_4.A_5$ (as 24T576) $0$ $0$
2.3775.120.b.b$2$ $ 5^{2} \cdot 151 $ 24.4.940350029043078386535797119140625.2 $C_4.A_5$ (as 24T576) $0$ $0$
2.3775.120.b.c$2$ $ 5^{2} \cdot 151 $ 24.4.940350029043078386535797119140625.2 $C_4.A_5$ (as 24T576) $0$ $0$
2.3775.120.b.d$2$ $ 5^{2} \cdot 151 $ 24.4.940350029043078386535797119140625.2 $C_4.A_5$ (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.b.a$4$ $ 5^{2} \cdot 151^{2}$ 24.4.940350029043078386535797119140625.2 $C_4.A_5$ (as 24T576) $0$ $0$
4.570025.40t188.b.b$4$ $ 5^{2} \cdot 151^{2}$ 24.4.940350029043078386535797119140625.2 $C_4.A_5$ (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.b.a$6$ $ 5^{4} \cdot 151^{3}$ 24.4.940350029043078386535797119140625.2 $C_4.A_5$ (as 24T576) $0$ $0$
* 6.2151844375.24t576.b.b$6$ $ 5^{4} \cdot 151^{3}$ 24.4.940350029043078386535797119140625.2 $C_4.A_5$ (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 *.