Properties

Label 24.4.374...536.1
Degree $24$
Signature $[4, 10]$
Discriminant $3.740\times 10^{33}$
Root discriminant $25.05$
Ramified primes $2, 751$
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 - 4*x^23 + 18*x^22 - 59*x^21 + 167*x^20 - 420*x^19 + 919*x^18 - 1828*x^17 + 3235*x^16 - 4879*x^15 + 6403*x^14 - 6415*x^13 + 2571*x^12 + 6076*x^11 - 21326*x^10 + 40942*x^9 - 60476*x^8 + 73984*x^7 - 73764*x^6 + 61224*x^5 - 43696*x^4 + 17928*x^3 - 3216*x^2 + 720*x + 16)
 
gp: K = bnfinit(x^24 - 4*x^23 + 18*x^22 - 59*x^21 + 167*x^20 - 420*x^19 + 919*x^18 - 1828*x^17 + 3235*x^16 - 4879*x^15 + 6403*x^14 - 6415*x^13 + 2571*x^12 + 6076*x^11 - 21326*x^10 + 40942*x^9 - 60476*x^8 + 73984*x^7 - 73764*x^6 + 61224*x^5 - 43696*x^4 + 17928*x^3 - 3216*x^2 + 720*x + 16, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![16, 720, -3216, 17928, -43696, 61224, -73764, 73984, -60476, 40942, -21326, 6076, 2571, -6415, 6403, -4879, 3235, -1828, 919, -420, 167, -59, 18, -4, 1]);
 

\( x^{24} - 4 x^{23} + 18 x^{22} - 59 x^{21} + 167 x^{20} - 420 x^{19} + 919 x^{18} - 1828 x^{17} + 3235 x^{16} - 4879 x^{15} + 6403 x^{14} - 6415 x^{13} + 2571 x^{12} + 6076 x^{11} - 21326 x^{10} + 40942 x^{9} - 60476 x^{8} + 73984 x^{7} - 73764 x^{6} + 61224 x^{5} - 43696 x^{4} + 17928 x^{3} - 3216 x^{2} + 720 x + 16 \)

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:  \(3740066297213363461879419371585536\)\(\medspace = 2^{16}\cdot 751^{10}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $25.05$
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:  $2, 751$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{12} - \frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{16} - \frac{1}{2} a^{13} - \frac{1}{2} a^{12} - \frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{17} - \frac{1}{2} a^{14} - \frac{1}{2} a^{13} - \frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5}$, $\frac{1}{4} a^{18} - \frac{1}{4} a^{15} - \frac{1}{4} a^{14} - \frac{1}{2} a^{13} - \frac{1}{4} a^{12} - \frac{1}{2} a^{11} + \frac{1}{4} a^{10} - \frac{1}{4} a^{9} + \frac{1}{4} a^{8} + \frac{1}{4} a^{7} - \frac{1}{4} a^{6} - \frac{1}{2} a^{4}$, $\frac{1}{4} a^{19} - \frac{1}{4} a^{16} - \frac{1}{4} a^{15} - \frac{1}{2} a^{14} - \frac{1}{4} a^{13} - \frac{1}{2} a^{12} + \frac{1}{4} a^{11} - \frac{1}{4} a^{10} + \frac{1}{4} a^{9} + \frac{1}{4} a^{8} - \frac{1}{4} a^{7} - \frac{1}{2} a^{5}$, $\frac{1}{4} a^{20} - \frac{1}{4} a^{17} - \frac{1}{4} a^{16} - \frac{1}{4} a^{14} - \frac{1}{2} a^{13} - \frac{1}{4} a^{12} + \frac{1}{4} a^{11} + \frac{1}{4} a^{10} - \frac{1}{4} a^{9} - \frac{1}{4} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{8} a^{21} - \frac{1}{8} a^{18} - \frac{1}{8} a^{17} - \frac{1}{4} a^{16} - \frac{1}{8} a^{15} - \frac{1}{4} a^{14} + \frac{1}{8} a^{13} + \frac{3}{8} a^{12} - \frac{3}{8} a^{11} - \frac{3}{8} a^{10} - \frac{1}{8} a^{9} - \frac{1}{4} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3}$, $\frac{1}{8} a^{22} - \frac{1}{8} a^{19} - \frac{1}{8} a^{18} - \frac{1}{4} a^{17} - \frac{1}{8} a^{16} - \frac{1}{4} a^{15} + \frac{1}{8} a^{14} + \frac{3}{8} a^{13} - \frac{3}{8} a^{12} - \frac{3}{8} a^{11} - \frac{1}{8} a^{10} - \frac{1}{4} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4}$, $\frac{1}{2762642112619498844741021572266061882880102264} a^{23} + \frac{102865210062477094592057378457622067818337619}{2762642112619498844741021572266061882880102264} a^{22} - \frac{72784612019316993468445258619733799172717917}{1381321056309749422370510786133030941440051132} a^{21} - \frac{10319580684129328130157376091820749477777543}{2762642112619498844741021572266061882880102264} a^{20} + \frac{54533813156942888003910088703327565243984911}{1381321056309749422370510786133030941440051132} a^{19} - \frac{232182670969798660690718238723443078676386481}{2762642112619498844741021572266061882880102264} a^{18} - \frac{647200135340064015653339125919807194472037687}{2762642112619498844741021572266061882880102264} a^{17} - \frac{661569299294629709856161380409347344074483881}{2762642112619498844741021572266061882880102264} a^{16} - \frac{433963421338042819378544149075442334716814155}{2762642112619498844741021572266061882880102264} a^{15} - \frac{55342058120846603316670611513958675882920175}{1381321056309749422370510786133030941440051132} a^{14} - \frac{264359030824412606020927881800642154420823525}{1381321056309749422370510786133030941440051132} a^{13} + \frac{487084847205413150724220478230427326930164999}{1381321056309749422370510786133030941440051132} a^{12} + \frac{78494493164122393365853320870818324050631402}{345330264077437355592627696533257735360012783} a^{11} - \frac{34594886321815730046509714440973656147344559}{2762642112619498844741021572266061882880102264} a^{10} + \frac{327418357995899011603286115816591727326391101}{1381321056309749422370510786133030941440051132} a^{9} - \frac{34157864053957466030853890003542613316062171}{690660528154874711185255393066515470720025566} a^{8} - \frac{30596250132247113585002203778171467924004117}{690660528154874711185255393066515470720025566} a^{7} - \frac{391003731382428052782495865215450238276703913}{1381321056309749422370510786133030941440051132} a^{6} + \frac{44301329643000673828317540734085610973141013}{690660528154874711185255393066515470720025566} a^{5} + \frac{103690657190438316177804299235857303796925169}{690660528154874711185255393066515470720025566} a^{4} - \frac{15666242017826402187312414793133057717883197}{345330264077437355592627696533257735360012783} a^{3} + \frac{30850618418593753378145816483129991038160498}{345330264077437355592627696533257735360012783} a^{2} + \frac{79965225970436942529848038639275127884897781}{345330264077437355592627696533257735360012783} a - \frac{106855904929287352993098074372431056616271096}{345330264077437355592627696533257735360012783}$

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:  \( 12097538.188247636 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{4}\cdot(2\pi)^{10}\cdot 12097538.188247636 \cdot 1}{2\sqrt{3740066297213363461879419371585536}}\approx 0.151756052800867$ (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

6.2.9024016.1, 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: 24.4.3740066297213363461879419371585536.2

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R ${\href{/LocalNumberField/3.12.0.1}{12} }^{2}$ ${\href{/LocalNumberField/5.6.0.1}{6} }^{4}$ $20{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }$ $20{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }$ ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.12.0.1}{12} }^{2}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{6}$ ${\href{/LocalNumberField/29.12.0.1}{12} }^{2}$ $20{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{6}$ $20{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }$ ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{6}$

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}$
751Data not computed

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
1.751.2t1.a.a$1$ $ 751 $ \(\Q(\sqrt{-751}) \) $C_2$ (as 2T1) $1$ $-1$
2.3004.120.b.a$2$ $ 2^{2} \cdot 751 $ 24.4.3740066297213363461879419371585536.1 $C_4.A_5$ (as 24T576) $0$ $0$
2.3004.120.b.b$2$ $ 2^{2} \cdot 751 $ 24.4.3740066297213363461879419371585536.1 $C_4.A_5$ (as 24T576) $0$ $0$
2.3004.120.b.c$2$ $ 2^{2} \cdot 751 $ 24.4.3740066297213363461879419371585536.1 $C_4.A_5$ (as 24T576) $0$ $0$
2.3004.120.b.d$2$ $ 2^{2} \cdot 751 $ 24.4.3740066297213363461879419371585536.1 $C_4.A_5$ (as 24T576) $0$ $0$
* 3.3004.12t76.a.a$3$ $ 2^{2} \cdot 751 $ 10.0.3822255090060016.1 $A_5\times C_2$ (as 10T11) $1$ $1$
* 3.3004.12t76.a.b$3$ $ 2^{2} \cdot 751 $ 10.0.3822255090060016.1 $A_5\times C_2$ (as 10T11) $1$ $1$
3.2256004.12t33.a.a$3$ $ 2^{2} \cdot 751^{2}$ 5.1.2256004.1 $A_5$ (as 5T4) $1$ $-1$
3.2256004.12t33.a.b$3$ $ 2^{2} \cdot 751^{2}$ 5.1.2256004.1 $A_5$ (as 5T4) $1$ $-1$
4.2256004.10t11.a.a$4$ $ 2^{2} \cdot 751^{2}$ 10.0.3822255090060016.1 $A_5\times C_2$ (as 10T11) $1$ $0$
4.2256004.5t4.a.a$4$ $ 2^{2} \cdot 751^{2}$ 5.1.2256004.1 $A_5$ (as 5T4) $1$ $0$
4.2256004.40t188.b.a$4$ $ 2^{2} \cdot 751^{2}$ 24.4.3740066297213363461879419371585536.1 $C_4.A_5$ (as 24T576) $0$ $0$
4.2256004.40t188.b.b$4$ $ 2^{2} \cdot 751^{2}$ 24.4.3740066297213363461879419371585536.1 $C_4.A_5$ (as 24T576) $0$ $0$
5.6777036016.12t75.a.a$5$ $ 2^{4} \cdot 751^{3}$ 10.0.3822255090060016.1 $A_5\times C_2$ (as 10T11) $1$ $-1$
* 5.9024016.6t12.a.a$5$ $ 2^{4} \cdot 751^{2}$ 5.1.2256004.1 $A_5$ (as 5T4) $1$ $1$
* 6.6777036016.24t576.b.a$6$ $ 2^{4} \cdot 751^{3}$ 24.4.3740066297213363461879419371585536.1 $C_4.A_5$ (as 24T576) $0$ $0$
* 6.6777036016.24t576.b.b$6$ $ 2^{4} \cdot 751^{3}$ 24.4.3740066297213363461879419371585536.1 $C_4.A_5$ (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 *.