Properties

Label 24.0.227...625.1
Degree $24$
Signature $[0, 12]$
Discriminant $2.273\times 10^{30}$
Root discriminant $18.40$
Ramified primes $3, 5, 7$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $C_{12}\times S_3$ (as 24T65)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^24 - 5*x^23 + 7*x^22 + 9*x^21 - 43*x^20 + 66*x^19 - 38*x^18 - 93*x^17 + 491*x^16 - 1291*x^15 + 2386*x^14 - 3246*x^13 + 3278*x^12 - 2354*x^11 + 1061*x^10 - 234*x^9 + 126*x^8 - 282*x^7 + 277*x^6 - 136*x^5 + 42*x^4 - 19*x^3 + 12*x^2 - 5*x + 1)
 
gp: K = bnfinit(x^24 - 5*x^23 + 7*x^22 + 9*x^21 - 43*x^20 + 66*x^19 - 38*x^18 - 93*x^17 + 491*x^16 - 1291*x^15 + 2386*x^14 - 3246*x^13 + 3278*x^12 - 2354*x^11 + 1061*x^10 - 234*x^9 + 126*x^8 - 282*x^7 + 277*x^6 - 136*x^5 + 42*x^4 - 19*x^3 + 12*x^2 - 5*x + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -5, 12, -19, 42, -136, 277, -282, 126, -234, 1061, -2354, 3278, -3246, 2386, -1291, 491, -93, -38, 66, -43, 9, 7, -5, 1]);
 

\(x^{24} - 5 x^{23} + 7 x^{22} + 9 x^{21} - 43 x^{20} + 66 x^{19} - 38 x^{18} - 93 x^{17} + 491 x^{16} - 1291 x^{15} + 2386 x^{14} - 3246 x^{13} + 3278 x^{12} - 2354 x^{11} + 1061 x^{10} - 234 x^{9} + 126 x^{8} - 282 x^{7} + 277 x^{6} - 136 x^{5} + 42 x^{4} - 19 x^{3} + 12 x^{2} - 5 x + 1\)  Toggle raw display

sage: K.defining_polynomial()
 
gp: K.pol
 
magma: DefiningPolynomial(K);
 

Invariants

Degree:  $24$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[0, 12]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(2272880662621757205963134765625\)\(\medspace = 3^{16}\cdot 5^{18}\cdot 7^{12}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $18.40$
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:  $3, 5, 7$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Aut(K/\Q)|$:  $12$
This field is not Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $\frac{1}{33} a^{20} - \frac{10}{33} a^{19} - \frac{1}{33} a^{18} + \frac{1}{11} a^{17} - \frac{1}{33} a^{16} - \frac{3}{11} a^{13} + \frac{4}{11} a^{12} - \frac{4}{33} a^{11} - \frac{5}{33} a^{9} + \frac{3}{11} a^{8} - \frac{14}{33} a^{7} - \frac{2}{33} a^{5} - \frac{8}{33} a^{4} - \frac{4}{33} a^{3} - \frac{4}{33} a^{2} + \frac{1}{33} a - \frac{8}{33}$, $\frac{1}{33} a^{21} - \frac{2}{33} a^{19} - \frac{7}{33} a^{18} - \frac{4}{33} a^{17} - \frac{10}{33} a^{16} - \frac{3}{11} a^{14} - \frac{4}{11} a^{13} - \frac{16}{33} a^{12} - \frac{7}{33} a^{11} - \frac{5}{33} a^{10} - \frac{8}{33} a^{9} + \frac{10}{33} a^{8} - \frac{8}{33} a^{7} - \frac{2}{33} a^{6} + \frac{5}{33} a^{5} + \frac{5}{11} a^{4} - \frac{1}{3} a^{3} - \frac{2}{11} a^{2} + \frac{2}{33} a - \frac{14}{33}$, $\frac{1}{101013} a^{22} - \frac{252}{33671} a^{21} - \frac{928}{101013} a^{20} + \frac{45646}{101013} a^{19} - \frac{17117}{101013} a^{18} + \frac{39869}{101013} a^{17} + \frac{1853}{101013} a^{16} + \frac{3682}{33671} a^{15} + \frac{13869}{33671} a^{14} + \frac{527}{101013} a^{13} + \frac{44834}{101013} a^{12} + \frac{11148}{33671} a^{11} - \frac{44540}{101013} a^{10} + \frac{22370}{101013} a^{9} - \frac{29960}{101013} a^{8} + \frac{9902}{101013} a^{7} + \frac{20591}{101013} a^{6} - \frac{40193}{101013} a^{5} + \frac{25526}{101013} a^{4} + \frac{44981}{101013} a^{3} + \frac{40648}{101013} a^{2} - \frac{47860}{101013} a - \frac{36260}{101013}$, $\frac{1}{5263958949618330357} a^{23} + \frac{446888716732}{1754652983206110119} a^{22} - \frac{53471461417170409}{5263958949618330357} a^{21} + \frac{38908752377430407}{5263958949618330357} a^{20} - \frac{184771365167908300}{1754652983206110119} a^{19} + \frac{2543216934336938320}{5263958949618330357} a^{18} + \frac{655459795700444264}{5263958949618330357} a^{17} + \frac{1363593263966905535}{5263958949618330357} a^{16} + \frac{591957268213237827}{1754652983206110119} a^{15} - \frac{1560066643166320261}{5263958949618330357} a^{14} + \frac{406760728184272034}{5263958949618330357} a^{13} - \frac{28979148637427185}{1754652983206110119} a^{12} - \frac{57357385259033900}{1754652983206110119} a^{11} + \frac{2285119256973322268}{5263958949618330357} a^{10} + \frac{242911682527489040}{5263958949618330357} a^{9} - \frac{2006564879839306387}{5263958949618330357} a^{8} - \frac{330885627906896803}{1754652983206110119} a^{7} + \frac{11173687806447425}{478541722692575487} a^{6} - \frac{142937571897312063}{1754652983206110119} a^{5} + \frac{152900597827733354}{1754652983206110119} a^{4} - \frac{563907950687164742}{1754652983206110119} a^{3} + \frac{853990555216872748}{5263958949618330357} a^{2} - \frac{1978962535552469923}{5263958949618330357} a + \frac{1398125949338705584}{5263958949618330357}$  Toggle raw display

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:  $11$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( -\frac{420542191288575801}{159513907564191829} a^{23} + \frac{1884319946847640178}{159513907564191829} a^{22} - \frac{1987664220605686358}{159513907564191829} a^{21} - \frac{4724789255142518635}{159513907564191829} a^{20} + \frac{1415299208988855822}{14501264324017439} a^{19} - \frac{19989949241358318038}{159513907564191829} a^{18} + \frac{6345062289037582560}{159513907564191829} a^{17} + \frac{41753912532463893142}{159513907564191829} a^{16} - \frac{185047803075614205948}{159513907564191829} a^{15} + \frac{449307406248197469246}{159513907564191829} a^{14} - \frac{779026316847090052525}{159513907564191829} a^{13} + \frac{980140157055206915327}{159513907564191829} a^{12} - \frac{899609150283943802129}{159513907564191829} a^{11} + \frac{554332968923387298268}{159513907564191829} a^{10} - \frac{179061393592224985443}{159513907564191829} a^{9} + \frac{9363005600615489148}{159513907564191829} a^{8} - \frac{41338870326134142235}{159513907564191829} a^{7} + \frac{92319816594770419231}{159513907564191829} a^{6} - \frac{70582448544126280591}{159513907564191829} a^{5} + \frac{24187758569682608476}{159513907564191829} a^{4} - \frac{5873819901447676869}{159513907564191829} a^{3} + \frac{4251476607857845654}{159513907564191829} a^{2} - \frac{2628319026083807795}{159513907564191829} a + \frac{812999923205803524}{159513907564191829} \) (order $10$)  Toggle raw display
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:  \( 1519657.3038774955 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{0}\cdot(2\pi)^{12}\cdot 1519657.3038774955 \cdot 1}{10\sqrt{2272880662621757205963134765625}}\approx 0.381606501439494$ (assuming GRH)

Galois group

$C_{12}\times S_3$ (as 24T65):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A solvable group of order 72
The 36 conjugacy class representatives for $C_{12}\times S_3$
Character table for $C_{12}\times S_3$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{-7}) \), \(\Q(\sqrt{-35}) \), \(\Q(\zeta_{5})\), 4.4.6125.1, \(\Q(\sqrt{5}, \sqrt{-7})\), 6.0.3472875.1, 8.0.37515625.1, 12.0.12060860765625.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 36 siblings: data not computed

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/LocalNumberField/2.12.0.1}{12} }^{2}$ R R R ${\href{/LocalNumberField/11.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{12}$ ${\href{/LocalNumberField/13.12.0.1}{12} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/17.12.0.1}{12} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{12}$ ${\href{/LocalNumberField/23.12.0.1}{12} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/43.12.0.1}{12} }^{2}$ ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{6}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{4}$

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
$3$3.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
3.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
3.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
3.12.16.14$x^{12} + 72 x^{11} - 36 x^{10} + 108 x^{9} - 108 x^{8} + 54 x^{7} + 72 x^{6} - 81 x^{5} - 81 x^{4} - 81 x^{3} + 81 x^{2} - 81$$3$$4$$16$$C_{12}$$[2]^{4}$
5Data not computed
7Data not computed

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
* 1.7.2t1.a.a$1$ $ 7 $ \(\Q(\sqrt{-7}) \) $C_2$ (as 2T1) $1$ $-1$
* 1.35.2t1.a.a$1$ $ 5 \cdot 7 $ \(\Q(\sqrt{-35}) \) $C_2$ (as 2T1) $1$ $-1$
* 1.5.2t1.a.a$1$ $ 5 $ \(\Q(\sqrt{5}) \) $C_2$ (as 2T1) $1$ $1$
1.315.6t1.h.a$1$ $ 3^{2} \cdot 5 \cdot 7 $ 6.0.281302875.3 $C_6$ (as 6T1) $0$ $-1$
1.315.6t1.h.b$1$ $ 3^{2} \cdot 5 \cdot 7 $ 6.0.281302875.3 $C_6$ (as 6T1) $0$ $-1$
1.63.6t1.c.a$1$ $ 3^{2} \cdot 7 $ 6.0.2250423.1 $C_6$ (as 6T1) $0$ $-1$
1.9.3t1.a.a$1$ $ 3^{2}$ \(\Q(\zeta_{9})^+\) $C_3$ (as 3T1) $0$ $1$
1.9.3t1.a.b$1$ $ 3^{2}$ \(\Q(\zeta_{9})^+\) $C_3$ (as 3T1) $0$ $1$
1.45.6t1.a.a$1$ $ 3^{2} \cdot 5 $ 6.6.820125.1 $C_6$ (as 6T1) $0$ $1$
1.45.6t1.a.b$1$ $ 3^{2} \cdot 5 $ 6.6.820125.1 $C_6$ (as 6T1) $0$ $1$
1.63.6t1.c.b$1$ $ 3^{2} \cdot 7 $ 6.0.2250423.1 $C_6$ (as 6T1) $0$ $-1$
* 1.35.4t1.a.a$1$ $ 5 \cdot 7 $ 4.4.6125.1 $C_4$ (as 4T1) $0$ $1$
* 1.5.4t1.a.a$1$ $ 5 $ \(\Q(\zeta_{5})\) $C_4$ (as 4T1) $0$ $-1$
* 1.5.4t1.a.b$1$ $ 5 $ \(\Q(\zeta_{5})\) $C_4$ (as 4T1) $0$ $-1$
* 1.35.4t1.a.b$1$ $ 5 \cdot 7 $ 4.4.6125.1 $C_4$ (as 4T1) $0$ $1$
1.45.12t1.a.a$1$ $ 3^{2} \cdot 5 $ 12.0.84075626953125.1 $C_{12}$ (as 12T1) $0$ $-1$
1.45.12t1.a.b$1$ $ 3^{2} \cdot 5 $ 12.0.84075626953125.1 $C_{12}$ (as 12T1) $0$ $-1$
1.315.12t1.a.a$1$ $ 3^{2} \cdot 5 \cdot 7 $ 12.12.9891413435408203125.1 $C_{12}$ (as 12T1) $0$ $1$
1.315.12t1.a.b$1$ $ 3^{2} \cdot 5 \cdot 7 $ 12.12.9891413435408203125.1 $C_{12}$ (as 12T1) $0$ $1$
1.45.12t1.a.c$1$ $ 3^{2} \cdot 5 $ 12.0.84075626953125.1 $C_{12}$ (as 12T1) $0$ $-1$
1.315.12t1.a.c$1$ $ 3^{2} \cdot 5 \cdot 7 $ 12.12.9891413435408203125.1 $C_{12}$ (as 12T1) $0$ $1$
1.315.12t1.a.d$1$ $ 3^{2} \cdot 5 \cdot 7 $ 12.12.9891413435408203125.1 $C_{12}$ (as 12T1) $0$ $1$
1.45.12t1.a.d$1$ $ 3^{2} \cdot 5 $ 12.0.84075626953125.1 $C_{12}$ (as 12T1) $0$ $-1$
2.2835.3t2.a.a$2$ $ 3^{4} \cdot 5 \cdot 7 $ 3.1.2835.1 $S_3$ (as 3T2) $1$ $0$
2.2835.6t3.e.a$2$ $ 3^{4} \cdot 5 \cdot 7 $ 6.0.56260575.1 $D_{6}$ (as 6T3) $1$ $0$
* 2.315.12t18.a.a$2$ $ 3^{2} \cdot 5 \cdot 7 $ 12.0.12060860765625.1 $C_6\times S_3$ (as 12T18) $0$ $0$
* 2.315.12t18.a.b$2$ $ 3^{2} \cdot 5 \cdot 7 $ 12.0.12060860765625.1 $C_6\times S_3$ (as 12T18) $0$ $0$
* 2.315.6t5.a.a$2$ $ 3^{2} \cdot 5 \cdot 7 $ 6.0.3472875.1 $S_3\times C_3$ (as 6T5) $0$ $0$
* 2.315.6t5.a.b$2$ $ 3^{2} \cdot 5 \cdot 7 $ 6.0.3472875.1 $S_3\times C_3$ (as 6T5) $0$ $0$
2.14175.12t11.a.a$2$ $ 3^{4} \cdot 5^{2} \cdot 7 $ 12.0.201865580314453125.1 $S_3 \times C_4$ (as 12T11) $0$ $0$
2.14175.12t11.a.b$2$ $ 3^{4} \cdot 5^{2} \cdot 7 $ 12.0.201865580314453125.1 $S_3 \times C_4$ (as 12T11) $0$ $0$
* 2.1575.24t65.a.a$2$ $ 3^{2} \cdot 5^{2} \cdot 7 $ 24.0.2272880662621757205963134765625.1 $C_{12}\times S_3$ (as 24T65) $0$ $0$
* 2.1575.24t65.a.b$2$ $ 3^{2} \cdot 5^{2} \cdot 7 $ 24.0.2272880662621757205963134765625.1 $C_{12}\times S_3$ (as 24T65) $0$ $0$
* 2.1575.24t65.a.c$2$ $ 3^{2} \cdot 5^{2} \cdot 7 $ 24.0.2272880662621757205963134765625.1 $C_{12}\times S_3$ (as 24T65) $0$ $0$
* 2.1575.24t65.a.d$2$ $ 3^{2} \cdot 5^{2} \cdot 7 $ 24.0.2272880662621757205963134765625.1 $C_{12}\times S_3$ (as 24T65) $0$ $0$

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 *.