Properties

Label 24.4.765...384.2
Degree $24$
Signature $[4, 10]$
Discriminant $7.654\times 10^{33}$
Root discriminant $25.81$
Ramified primes $2, 3, 11$
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 + 6*x^22 - 19*x^21 + 39*x^20 - 30*x^19 - 9*x^18 - 45*x^17 + 93*x^16 + 204*x^15 - 201*x^14 - 129*x^13 - 793*x^12 + 738*x^11 + 141*x^10 + 1177*x^9 - 666*x^8 - 597*x^7 - 913*x^6 + 168*x^5 + 474*x^4 + 464*x^3 + 168*x^2 + 24*x - 4)
 
gp: K = bnfinit(x^24 - 3*x^23 + 6*x^22 - 19*x^21 + 39*x^20 - 30*x^19 - 9*x^18 - 45*x^17 + 93*x^16 + 204*x^15 - 201*x^14 - 129*x^13 - 793*x^12 + 738*x^11 + 141*x^10 + 1177*x^9 - 666*x^8 - 597*x^7 - 913*x^6 + 168*x^5 + 474*x^4 + 464*x^3 + 168*x^2 + 24*x - 4, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-4, 24, 168, 464, 474, 168, -913, -597, -666, 1177, 141, 738, -793, -129, -201, 204, 93, -45, -9, -30, 39, -19, 6, -3, 1]);
 

\( x^{24} - 3 x^{23} + 6 x^{22} - 19 x^{21} + 39 x^{20} - 30 x^{19} - 9 x^{18} - 45 x^{17} + 93 x^{16} + 204 x^{15} - 201 x^{14} - 129 x^{13} - 793 x^{12} + 738 x^{11} + 141 x^{10} + 1177 x^{9} - 666 x^{8} - 597 x^{7} - 913 x^{6} + 168 x^{5} + 474 x^{4} + 464 x^{3} + 168 x^{2} + 24 x - 4 \)

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:  \(7654476884289954721505578121232384\)\(\medspace = 2^{16}\cdot 3^{26}\cdot 11^{16}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $25.81$
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:  $2, 3, 11$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{11} - \frac{1}{2} a^{8} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{16} - \frac{1}{4} a^{14} - \frac{1}{4} a^{13} - \frac{1}{4} a^{11} - \frac{1}{4} a^{10} - \frac{1}{2} a^{9} + \frac{1}{4} a^{8} + \frac{1}{4} a^{7} + \frac{1}{4} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{17} - \frac{1}{4} a^{15} - \frac{1}{4} a^{14} - \frac{1}{4} a^{12} - \frac{1}{4} a^{11} - \frac{1}{2} a^{10} + \frac{1}{4} a^{9} + \frac{1}{4} a^{8} + \frac{1}{4} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{12} a^{18} + \frac{1}{12} a^{15} - \frac{1}{4} a^{14} + \frac{1}{12} a^{12} + \frac{1}{4} a^{11} - \frac{1}{2} a^{10} + \frac{1}{12} a^{9} + \frac{1}{4} a^{8} - \frac{1}{4} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{3}$, $\frac{1}{24} a^{19} + \frac{1}{24} a^{16} + \frac{1}{8} a^{15} - \frac{1}{4} a^{14} + \frac{1}{24} a^{13} + \frac{1}{8} a^{12} - \frac{1}{2} a^{11} - \frac{11}{24} a^{10} + \frac{1}{8} a^{9} - \frac{1}{4} a^{8} - \frac{1}{2} a^{7} + \frac{3}{8} a^{6} + \frac{1}{4} a^{4} - \frac{1}{2} a^{3} - \frac{1}{6} a - \frac{1}{2}$, $\frac{1}{24} a^{20} + \frac{1}{24} a^{17} - \frac{1}{8} a^{16} - \frac{1}{4} a^{15} - \frac{5}{24} a^{14} - \frac{1}{8} a^{13} + \frac{7}{24} a^{11} - \frac{1}{8} a^{10} - \frac{1}{4} a^{9} - \frac{1}{4} a^{8} - \frac{3}{8} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} + \frac{1}{3} a^{2} - \frac{1}{2} a$, $\frac{1}{48} a^{21} - \frac{1}{48} a^{19} + \frac{1}{48} a^{18} - \frac{1}{16} a^{17} - \frac{1}{48} a^{16} - \frac{1}{6} a^{15} + \frac{3}{16} a^{14} - \frac{7}{48} a^{13} + \frac{1}{12} a^{12} + \frac{5}{16} a^{11} + \frac{23}{48} a^{10} + \frac{1}{16} a^{9} - \frac{3}{16} a^{8} + \frac{1}{8} a^{7} - \frac{7}{16} a^{6} - \frac{3}{8} a^{5} - \frac{3}{8} a^{4} + \frac{1}{6} a^{3} + \frac{1}{12} a + \frac{1}{4}$, $\frac{1}{48} a^{22} - \frac{1}{48} a^{20} - \frac{1}{48} a^{19} + \frac{1}{48} a^{18} - \frac{1}{48} a^{17} + \frac{1}{24} a^{16} + \frac{7}{48} a^{15} + \frac{5}{48} a^{14} - \frac{5}{24} a^{13} - \frac{11}{48} a^{12} + \frac{23}{48} a^{11} - \frac{11}{48} a^{10} - \frac{11}{48} a^{9} + \frac{3}{8} a^{8} + \frac{5}{16} a^{7} - \frac{1}{4} a^{6} + \frac{1}{8} a^{5} - \frac{1}{12} a^{4} + \frac{1}{12} a^{2} + \frac{5}{12} a + \frac{1}{6}$, $\frac{1}{1827309298827628347823445856} a^{23} - \frac{1473817888801727544648093}{152275774902302362318620488} a^{22} + \frac{561732763673785791050725}{228413662353453543477930732} a^{21} + \frac{18766426272289680202899}{609103099609209449274481952} a^{20} + \frac{1711180052499363449541419}{456827324706907086955861464} a^{19} + \frac{1354392773755357364927629}{152275774902302362318620488} a^{18} - \frac{66089784628356298472352377}{609103099609209449274481952} a^{17} - \frac{902392813745836878754898}{57103415588363385869482683} a^{16} - \frac{29400971099169878176702333}{609103099609209449274481952} a^{15} + \frac{131466160437149537603220479}{609103099609209449274481952} a^{14} - \frac{6826950245498569514197235}{57103415588363385869482683} a^{13} + \frac{71099033323294796939377881}{609103099609209449274481952} a^{12} + \frac{309828025821753902644921297}{913654649413814173911722928} a^{11} + \frac{205668700153897323799937713}{913654649413814173911722928} a^{10} - \frac{98350626335332943114929607}{1827309298827628347823445856} a^{9} + \frac{138314270426675456649363415}{304551549804604724637240976} a^{8} - \frac{27271659641953150581831749}{152275774902302362318620488} a^{7} + \frac{189062124276697438432606919}{609103099609209449274481952} a^{6} - \frac{360345345202777521763306397}{913654649413814173911722928} a^{5} - \frac{86552670562055707660426831}{304551549804604724637240976} a^{4} - \frac{27531235076728595814907139}{456827324706907086955861464} a^{3} + \frac{120072835544684775292226263}{456827324706907086955861464} a^{2} + \frac{127966509442727737961843197}{456827324706907086955861464} a + \frac{222638534728156948463669519}{456827324706907086955861464}$

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:  \( 494355842.3116425 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{4}\cdot(2\pi)^{10}\cdot 494355842.3116425 \cdot 1}{2\sqrt{7654476884289954721505578121232384}}\approx 4.33481516125310$ (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

6.2.170772624.2, 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: 24.4.7654476884289954721505578121232384.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R R $20{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{4}$ R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ $20{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{8}$ $20{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }$ ${\href{/LocalNumberField/29.12.0.1}{12} }^{2}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{6}$ ${\href{/LocalNumberField/41.12.0.1}{12} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{6}$ $20{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }$ ${\href{/LocalNumberField/53.12.0.1}{12} }^{2}$ ${\href{/LocalNumberField/59.12.0.1}{12} }^{2}$

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$2.3.2.1$x^{3} - 2$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
2.3.2.1$x^{3} - 2$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
2.3.2.1$x^{3} - 2$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
2.3.2.1$x^{3} - 2$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
$3$3.12.12.5$x^{12} + 33 x^{11} - 63 x^{10} - 36 x^{9} - 90 x^{8} - 54 x^{7} - 54 x^{6} - 108 x^{4} - 27 x^{3} - 81 x^{2} + 81 x - 81$$3$$4$$12$$S_3 \times C_4$$[3/2]_{2}^{4}$
3.12.14.9$x^{12} + 6 x^{11} + 12 x^{10} - 3 x^{9} + 12 x^{6} - 9 x^{5} + 9 x^{4} - 9 x^{3} - 9 x^{2} - 9$$6$$2$$14$$S_3 \times C_4$$[3/2]_{2}^{4}$
11Data not computed

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
1.3.2t1.a.a$1$ $ 3 $ \(\Q(\sqrt{-3}) \) $C_2$ (as 2T1) $1$ $-1$
2.13068.120.a.a$2$ $ 2^{2} \cdot 3^{3} \cdot 11^{2}$ 24.4.7654476884289954721505578121232384.2 $C_4.A_5$ (as 24T576) $0$ $0$
2.13068.120.a.b$2$ $ 2^{2} \cdot 3^{3} \cdot 11^{2}$ 24.4.7654476884289954721505578121232384.2 $C_4.A_5$ (as 24T576) $0$ $0$
2.13068.120.a.c$2$ $ 2^{2} \cdot 3^{3} \cdot 11^{2}$ 24.4.7654476884289954721505578121232384.2 $C_4.A_5$ (as 24T576) $0$ $0$
2.13068.120.a.d$2$ $ 2^{2} \cdot 3^{3} \cdot 11^{2}$ 24.4.7654476884289954721505578121232384.2 $C_4.A_5$ (as 24T576) $0$ $0$
* 3.13068.12t76.a.a$3$ $ 2^{2} \cdot 3^{3} \cdot 11^{2}$ 10.0.67507613675568.1 $A_5\times C_2$ (as 10T11) $1$ $1$
* 3.13068.12t76.a.b$3$ $ 2^{2} \cdot 3^{3} \cdot 11^{2}$ 10.0.67507613675568.1 $A_5\times C_2$ (as 10T11) $1$ $1$
3.39204.12t33.a.a$3$ $ 2^{2} \cdot 3^{4} \cdot 11^{2}$ 5.1.4743684.1 $A_5$ (as 5T4) $1$ $-1$
3.39204.12t33.a.b$3$ $ 2^{2} \cdot 3^{4} \cdot 11^{2}$ 5.1.4743684.1 $A_5$ (as 5T4) $1$ $-1$
4.4743684.10t11.a.a$4$ $ 2^{2} \cdot 3^{4} \cdot 11^{4}$ 10.0.67507613675568.1 $A_5\times C_2$ (as 10T11) $1$ $0$
4.4743684.5t4.a.a$4$ $ 2^{2} \cdot 3^{4} \cdot 11^{4}$ 5.1.4743684.1 $A_5$ (as 5T4) $1$ $0$
4.4743684.40t188.a.a$4$ $ 2^{2} \cdot 3^{4} \cdot 11^{4}$ 24.4.7654476884289954721505578121232384.2 $C_4.A_5$ (as 24T576) $0$ $0$
4.4743684.40t188.a.b$4$ $ 2^{2} \cdot 3^{4} \cdot 11^{4}$ 24.4.7654476884289954721505578121232384.2 $C_4.A_5$ (as 24T576) $0$ $0$
5.512317872.12t75.a.a$5$ $ 2^{4} \cdot 3^{7} \cdot 11^{4}$ 10.0.67507613675568.1 $A_5\times C_2$ (as 10T11) $1$ $-1$
* 5.170772624.6t12.a.a$5$ $ 2^{4} \cdot 3^{6} \cdot 11^{4}$ 5.1.4743684.1 $A_5$ (as 5T4) $1$ $1$
* 6.512317872.24t576.a.a$6$ $ 2^{4} \cdot 3^{7} \cdot 11^{4}$ 24.4.7654476884289954721505578121232384.2 $C_4.A_5$ (as 24T576) $0$ $0$
* 6.512317872.24t576.a.b$6$ $ 2^{4} \cdot 3^{7} \cdot 11^{4}$ 24.4.7654476884289954721505578121232384.2 $C_4.A_5$ (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 *.