Properties

Label 24.4.374...536.2
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 - 3*x^23 + x^22 + 10*x^21 - 43*x^20 + 48*x^19 + 120*x^18 - 317*x^17 + 184*x^16 + 607*x^15 - 1095*x^14 - 247*x^13 + 1345*x^12 - 781*x^11 + 2081*x^10 + 2799*x^9 - 6866*x^8 - 6853*x^7 + 2370*x^6 + 3678*x^5 + 629*x^4 - 178*x^3 - 33*x^2 - x + 1)
 
gp: K = bnfinit(x^24 - 3*x^23 + x^22 + 10*x^21 - 43*x^20 + 48*x^19 + 120*x^18 - 317*x^17 + 184*x^16 + 607*x^15 - 1095*x^14 - 247*x^13 + 1345*x^12 - 781*x^11 + 2081*x^10 + 2799*x^9 - 6866*x^8 - 6853*x^7 + 2370*x^6 + 3678*x^5 + 629*x^4 - 178*x^3 - 33*x^2 - x + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -1, -33, -178, 629, 3678, 2370, -6853, -6866, 2799, 2081, -781, 1345, -247, -1095, 607, 184, -317, 120, 48, -43, 10, 1, -3, 1]);
 

\( x^{24} - 3 x^{23} + x^{22} + 10 x^{21} - 43 x^{20} + 48 x^{19} + 120 x^{18} - 317 x^{17} + 184 x^{16} + 607 x^{15} - 1095 x^{14} - 247 x^{13} + 1345 x^{12} - 781 x^{11} + 2081 x^{10} + 2799 x^{9} - 6866 x^{8} - 6853 x^{7} + 2370 x^{6} + 3678 x^{5} + 629 x^{4} - 178 x^{3} - 33 x^{2} - x + 1 \)

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:  \(3740066297213363461879419371585536\)\(\medspace = 2^{16}\cdot 751^{10}\)
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $\frac{1}{871506839132812529726775283009574219585257} a^{23} + \frac{171349164318814915081720774398295885622170}{871506839132812529726775283009574219585257} a^{22} + \frac{83270601910518521360178582359606933066783}{871506839132812529726775283009574219585257} a^{21} + \frac{185015981251216393142884063905964874697961}{871506839132812529726775283009574219585257} a^{20} + \frac{9076639175150913221007174263243652953216}{871506839132812529726775283009574219585257} a^{19} + \frac{80025982182995665332447003444446976090521}{871506839132812529726775283009574219585257} a^{18} - \frac{205481072367377663271054797899208460403714}{871506839132812529726775283009574219585257} a^{17} - \frac{428183597176976051648911200836180937956704}{871506839132812529726775283009574219585257} a^{16} + \frac{300897489862204039981014453681235362235168}{871506839132812529726775283009574219585257} a^{15} + \frac{426800599478135888373490782663225575199296}{871506839132812529726775283009574219585257} a^{14} + \frac{7266282081408564557705182194639835670410}{871506839132812529726775283009574219585257} a^{13} + \frac{101754179932564798566464765826313980527519}{871506839132812529726775283009574219585257} a^{12} - \frac{53498626163714627898737042298275394686666}{871506839132812529726775283009574219585257} a^{11} - \frac{367429784336785238232136244306270993627034}{871506839132812529726775283009574219585257} a^{10} + \frac{44533466401942839897779842211832879526907}{871506839132812529726775283009574219585257} a^{9} - \frac{265234218130513700665364982073862066474520}{871506839132812529726775283009574219585257} a^{8} - \frac{117078681805163563805248329233932030121549}{871506839132812529726775283009574219585257} a^{7} - \frac{134414374085746639692689686770562179542547}{871506839132812529726775283009574219585257} a^{6} + \frac{57923845431592406422187819812529221021390}{871506839132812529726775283009574219585257} a^{5} - \frac{193905303783299388172948296483239537364248}{871506839132812529726775283009574219585257} a^{4} - \frac{333489578198479707884902696994215282588251}{871506839132812529726775283009574219585257} a^{3} - \frac{277101142282695556712522274231812511445882}{871506839132812529726775283009574219585257} a^{2} + \frac{49545594473227120004747254599600911362056}{871506839132812529726775283009574219585257} a + \frac{45820920425031178721306174078431534134836}{871506839132812529726775283009574219585257}$

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: 24.4.3740066297213363461879419371585536.1

Frobenius cycle types

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

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 Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ $x$ $C_1$ $1$ $1$
1.751.2t1.a.a$1$ $ 751 $ $x^{2} - x + 188$ $C_2$ (as 2T1) $1$ $-1$
2.3004.120.a.a$2$ $ 2^{2} \cdot 751 $ $x^{24} - 3 x^{23} + x^{22} + 10 x^{21} - 43 x^{20} + 48 x^{19} + 120 x^{18} - 317 x^{17} + 184 x^{16} + 607 x^{15} - 1095 x^{14} - 247 x^{13} + 1345 x^{12} - 781 x^{11} + 2081 x^{10} + 2799 x^{9} - 6866 x^{8} - 6853 x^{7} + 2370 x^{6} + 3678 x^{5} + 629 x^{4} - 178 x^{3} - 33 x^{2} - x + 1$ $C_4.A_5$ (as 24T576) $0$ $0$
2.3004.120.a.b$2$ $ 2^{2} \cdot 751 $ $x^{24} - 3 x^{23} + x^{22} + 10 x^{21} - 43 x^{20} + 48 x^{19} + 120 x^{18} - 317 x^{17} + 184 x^{16} + 607 x^{15} - 1095 x^{14} - 247 x^{13} + 1345 x^{12} - 781 x^{11} + 2081 x^{10} + 2799 x^{9} - 6866 x^{8} - 6853 x^{7} + 2370 x^{6} + 3678 x^{5} + 629 x^{4} - 178 x^{3} - 33 x^{2} - x + 1$ $C_4.A_5$ (as 24T576) $0$ $0$
2.3004.120.a.c$2$ $ 2^{2} \cdot 751 $ $x^{24} - 3 x^{23} + x^{22} + 10 x^{21} - 43 x^{20} + 48 x^{19} + 120 x^{18} - 317 x^{17} + 184 x^{16} + 607 x^{15} - 1095 x^{14} - 247 x^{13} + 1345 x^{12} - 781 x^{11} + 2081 x^{10} + 2799 x^{9} - 6866 x^{8} - 6853 x^{7} + 2370 x^{6} + 3678 x^{5} + 629 x^{4} - 178 x^{3} - 33 x^{2} - x + 1$ $C_4.A_5$ (as 24T576) $0$ $0$
2.3004.120.a.d$2$ $ 2^{2} \cdot 751 $ $x^{24} - 3 x^{23} + x^{22} + 10 x^{21} - 43 x^{20} + 48 x^{19} + 120 x^{18} - 317 x^{17} + 184 x^{16} + 607 x^{15} - 1095 x^{14} - 247 x^{13} + 1345 x^{12} - 781 x^{11} + 2081 x^{10} + 2799 x^{9} - 6866 x^{8} - 6853 x^{7} + 2370 x^{6} + 3678 x^{5} + 629 x^{4} - 178 x^{3} - 33 x^{2} - x + 1$ $C_4.A_5$ (as 24T576) $0$ $0$
* 3.3004.12t76.a.a$3$ $ 2^{2} \cdot 751 $ $x^{10} - 2 x^{9} - 5 x^{8} + 17 x^{7} - 75 x^{5} + 252 x^{4} - 620 x^{3} + 532 x^{2} - 40 x + 400$ $A_5\times C_2$ (as 10T11) $1$ $1$
* 3.3004.12t76.a.b$3$ $ 2^{2} \cdot 751 $ $x^{10} - 2 x^{9} - 5 x^{8} + 17 x^{7} - 75 x^{5} + 252 x^{4} - 620 x^{3} + 532 x^{2} - 40 x + 400$ $A_5\times C_2$ (as 10T11) $1$ $1$
3.2256004.12t33.a.a$3$ $ 2^{2} \cdot 751^{2}$ $x^{5} - 2 x^{4} - x^{3} - 3 x^{2} + 2 x - 3$ $A_5$ (as 5T4) $1$ $-1$
3.2256004.12t33.a.b$3$ $ 2^{2} \cdot 751^{2}$ $x^{5} - 2 x^{4} - x^{3} - 3 x^{2} + 2 x - 3$ $A_5$ (as 5T4) $1$ $-1$
4.2256004.10t11.a.a$4$ $ 2^{2} \cdot 751^{2}$ $x^{10} - 2 x^{9} - 5 x^{8} + 17 x^{7} - 75 x^{5} + 252 x^{4} - 620 x^{3} + 532 x^{2} - 40 x + 400$ $A_5\times C_2$ (as 10T11) $1$ $0$
4.2256004.5t4.a.a$4$ $ 2^{2} \cdot 751^{2}$ $x^{5} - 2 x^{4} - x^{3} - 3 x^{2} + 2 x - 3$ $A_5$ (as 5T4) $1$ $0$
4.2256004.40t188.a.a$4$ $ 2^{2} \cdot 751^{2}$ $x^{24} - 3 x^{23} + x^{22} + 10 x^{21} - 43 x^{20} + 48 x^{19} + 120 x^{18} - 317 x^{17} + 184 x^{16} + 607 x^{15} - 1095 x^{14} - 247 x^{13} + 1345 x^{12} - 781 x^{11} + 2081 x^{10} + 2799 x^{9} - 6866 x^{8} - 6853 x^{7} + 2370 x^{6} + 3678 x^{5} + 629 x^{4} - 178 x^{3} - 33 x^{2} - x + 1$ $C_4.A_5$ (as 24T576) $0$ $0$
4.2256004.40t188.a.b$4$ $ 2^{2} \cdot 751^{2}$ $x^{24} - 3 x^{23} + x^{22} + 10 x^{21} - 43 x^{20} + 48 x^{19} + 120 x^{18} - 317 x^{17} + 184 x^{16} + 607 x^{15} - 1095 x^{14} - 247 x^{13} + 1345 x^{12} - 781 x^{11} + 2081 x^{10} + 2799 x^{9} - 6866 x^{8} - 6853 x^{7} + 2370 x^{6} + 3678 x^{5} + 629 x^{4} - 178 x^{3} - 33 x^{2} - x + 1$ $C_4.A_5$ (as 24T576) $0$ $0$
5.6777036016.12t75.a.a$5$ $ 2^{4} \cdot 751^{3}$ $x^{10} - 2 x^{9} - 5 x^{8} + 17 x^{7} - 75 x^{5} + 252 x^{4} - 620 x^{3} + 532 x^{2} - 40 x + 400$ $A_5\times C_2$ (as 10T11) $1$ $-1$
* 5.9024016.6t12.a.a$5$ $ 2^{4} \cdot 751^{2}$ $x^{5} - 2 x^{4} - x^{3} - 3 x^{2} + 2 x - 3$ $A_5$ (as 5T4) $1$ $1$
* 6.6777036016.24t576.a.a$6$ $ 2^{4} \cdot 751^{3}$ $x^{24} - 3 x^{23} + x^{22} + 10 x^{21} - 43 x^{20} + 48 x^{19} + 120 x^{18} - 317 x^{17} + 184 x^{16} + 607 x^{15} - 1095 x^{14} - 247 x^{13} + 1345 x^{12} - 781 x^{11} + 2081 x^{10} + 2799 x^{9} - 6866 x^{8} - 6853 x^{7} + 2370 x^{6} + 3678 x^{5} + 629 x^{4} - 178 x^{3} - 33 x^{2} - x + 1$ $C_4.A_5$ (as 24T576) $0$ $0$
* 6.6777036016.24t576.a.b$6$ $ 2^{4} \cdot 751^{3}$ $x^{24} - 3 x^{23} + x^{22} + 10 x^{21} - 43 x^{20} + 48 x^{19} + 120 x^{18} - 317 x^{17} + 184 x^{16} + 607 x^{15} - 1095 x^{14} - 247 x^{13} + 1345 x^{12} - 781 x^{11} + 2081 x^{10} + 2799 x^{9} - 6866 x^{8} - 6853 x^{7} + 2370 x^{6} + 3678 x^{5} + 629 x^{4} - 178 x^{3} - 33 x^{2} - x + 1$ $C_4.A_5$ (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 *.