Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^24 - 6*x^23 + 21*x^22 - 54*x^21 + 107*x^20 - 178*x^19 + 265*x^18 - 376*x^17 + 537*x^16 - 744*x^15 + 966*x^14 - 1140*x^13 + 1221*x^12 - 1228*x^11 + 1176*x^10 - 1074*x^9 + 914*x^8 - 696*x^7 + 471*x^6 - 280*x^5 + 140*x^4 - 58*x^3 + 21*x^2 - 6*x + 1)
gp: K = bnfinit(x^24 - 6*x^23 + 21*x^22 - 54*x^21 + 107*x^20 - 178*x^19 + 265*x^18 - 376*x^17 + 537*x^16 - 744*x^15 + 966*x^14 - 1140*x^13 + 1221*x^12 - 1228*x^11 + 1176*x^10 - 1074*x^9 + 914*x^8 - 696*x^7 + 471*x^6 - 280*x^5 + 140*x^4 - 58*x^3 + 21*x^2 - 6*x + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -6, 21, -58, 140, -280, 471, -696, 914, -1074, 1176, -1228, 1221, -1140, 966, -744, 537, -376, 265, -178, 107, -54, 21, -6, 1]);
\( x^{24} - 6 x^{23} + 21 x^{22} - 54 x^{21} + 107 x^{20} - 178 x^{19} + 265 x^{18} - 376 x^{17} + 537 x^{16} - 744 x^{15} + 966 x^{14} - 1140 x^{13} + 1221 x^{12} - 1228 x^{11} + 1176 x^{10} - 1074 x^{9} + 914 x^{8} - 696 x^{7} + 471 x^{6} - 280 x^{5} + 140 x^{4} - 58 x^{3} + 21 x^{2} - 6 x + 1 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
Invariants
| Degree: | $24$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
| |
| Signature: | $[0, 12]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
| |
| Discriminant: | \(368947264000000000000000000\)\(\medspace = 2^{24}\cdot 5^{18}\cdot 7^{8}\) | sage: K.disc()
gp: K.disc
magma: Discriminant(Integers(K));
| |
| Root discriminant: | $12.79$ | 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: | $2, 5, 7$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $\frac{1}{191679830229179} a^{23} + \frac{8422786950576}{191679830229179} a^{22} - \frac{22694145349783}{191679830229179} a^{21} - \frac{6161595013249}{191679830229179} a^{20} - \frac{73723363546300}{191679830229179} a^{19} + \frac{55063250317610}{191679830229179} a^{18} + \frac{33976018607284}{191679830229179} a^{17} - \frac{85199078093611}{191679830229179} a^{16} + \frac{33893792053320}{191679830229179} a^{15} + \frac{31808053967254}{191679830229179} a^{14} + \frac{54734987496610}{191679830229179} a^{13} + \frac{5199015835275}{191679830229179} a^{12} + \frac{59348984355966}{191679830229179} a^{11} - \frac{32238076066833}{191679830229179} a^{10} - \frac{79659112112203}{191679830229179} a^{9} + \frac{42917911640006}{191679830229179} a^{8} + \frac{15963632334858}{191679830229179} a^{7} - \frac{40481349346422}{191679830229179} a^{6} + \frac{21920184363788}{191679830229179} a^{5} - \frac{54317148824536}{191679830229179} a^{4} - \frac{64837609145563}{191679830229179} a^{3} - \frac{21523508422520}{191679830229179} a^{2} - \frac{69505981625860}{191679830229179} a + \frac{95071079886986}{191679830229179}$
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: | \( -\frac{246938601708342}{191679830229179} a^{23} + \frac{1390625883620859}{191679830229179} a^{22} - \frac{4690132406743003}{191679830229179} a^{21} + \frac{11693310212992961}{191679830229179} a^{20} - \frac{22407405942837677}{191679830229179} a^{19} + \frac{36436954373691654}{191679830229179} a^{18} - \frac{53417981694349363}{191679830229179} a^{17} + \frac{75464163619737171}{191679830229179} a^{16} - \frac{108114681538802938}{191679830229179} a^{15} + \frac{148486908248637245}{191679830229179} a^{14} - \frac{190461567438894078}{191679830229179} a^{13} + \frac{220408936121823028}{191679830229179} a^{12} - \frac{232075014107646222}{191679830229179} a^{11} + \frac{231190167248330130}{191679830229179} a^{10} - \frac{219003463204885278}{191679830229179} a^{9} + \frac{198179686228003322}{191679830229179} a^{8} - \frac{165442350017954611}{191679830229179} a^{7} + \frac{122617740183232305}{191679830229179} a^{6} - \frac{80797338352179268}{191679830229179} a^{5} + \frac{46132174266051938}{191679830229179} a^{4} - \frac{21969283656399143}{191679830229179} a^{3} + \frac{8687105857380701}{191679830229179} a^{2} - \frac{3018529100074687}{191679830229179} a + \frac{731375154086694}{191679830229179} \) (order $20$) | 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: | \( 15456.373586053287 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
|
Class number formula
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
| \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-5}) \), \(\Q(\sqrt{-1}) \), \(\Q(\zeta_{5})\), \(\Q(\zeta_{20})^+\), \(\Q(i, \sqrt{5})\), 6.0.392000.1, \(\Q(\zeta_{20})\), 12.0.153664000000.1 |
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 | R | ${\href{/LocalNumberField/3.12.0.1}{12} }{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }^{3}$ | R | R | ${\href{/LocalNumberField/11.6.0.1}{6} }^{4}$ | ${\href{/LocalNumberField/13.4.0.1}{4} }^{6}$ | ${\href{/LocalNumberField/17.12.0.1}{12} }^{2}$ | ${\href{/LocalNumberField/19.6.0.1}{6} }^{4}$ | ${\href{/LocalNumberField/23.12.0.1}{12} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }^{3}$ | ${\href{/LocalNumberField/29.6.0.1}{6} }^{4}$ | ${\href{/LocalNumberField/31.6.0.1}{6} }^{4}$ | ${\href{/LocalNumberField/37.12.0.1}{12} }^{2}$ | ${\href{/LocalNumberField/41.3.0.1}{3} }^{8}$ | ${\href{/LocalNumberField/43.12.0.1}{12} }^{2}$ | ${\href{/LocalNumberField/47.12.0.1}{12} }^{2}$ | ${\href{/LocalNumberField/53.12.0.1}{12} }^{2}$ | ${\href{/LocalNumberField/59.6.0.1}{6} }^{4}$ |
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
| $p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
|---|---|---|---|---|---|---|---|
| 2 | Data not computed | ||||||
| $5$ | 5.12.9.2 | $x^{12} - 10 x^{8} + 25 x^{4} - 500$ | $4$ | $3$ | $9$ | $C_{12}$ | $[\ ]_{4}^{3}$ |
| 5.12.9.2 | $x^{12} - 10 x^{8} + 25 x^{4} - 500$ | $4$ | $3$ | $9$ | $C_{12}$ | $[\ ]_{4}^{3}$ | |
| $7$ | 7.12.0.1 | $x^{12} + 3 x^{2} - 2 x + 3$ | $1$ | $12$ | $0$ | $C_{12}$ | $[\ ]^{12}$ |
| 7.12.8.3 | $x^{12} + 14 x^{9} + 539 x^{6} + 343 x^{3} + 60025$ | $3$ | $4$ | $8$ | $C_{12}$ | $[\ ]_{3}^{4}$ | |
Artin representations
| Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| * | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
| * | 1.4.2t1.a.a | $1$ | $ 2^{2}$ | \(\Q(\sqrt{-1}) \) | $C_2$ (as 2T1) | $1$ | $-1$ |
| * | 1.20.2t1.a.a | $1$ | $ 2^{2} \cdot 5 $ | \(\Q(\sqrt{-5}) \) | $C_2$ (as 2T1) | $1$ | $-1$ |
| * | 1.5.2t1.a.a | $1$ | $ 5 $ | \(\Q(\sqrt{5}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
| 1.140.6t1.b.a | $1$ | $ 2^{2} \cdot 5 \cdot 7 $ | 6.0.19208000.1 | $C_6$ (as 6T1) | $0$ | $-1$ | |
| 1.7.3t1.a.a | $1$ | $ 7 $ | \(\Q(\zeta_{7})^+\) | $C_3$ (as 3T1) | $0$ | $1$ | |
| 1.140.6t1.b.b | $1$ | $ 2^{2} \cdot 5 \cdot 7 $ | 6.0.19208000.1 | $C_6$ (as 6T1) | $0$ | $-1$ | |
| 1.28.6t1.a.a | $1$ | $ 2^{2} \cdot 7 $ | 6.0.153664.1 | $C_6$ (as 6T1) | $0$ | $-1$ | |
| 1.35.6t1.b.a | $1$ | $ 5 \cdot 7 $ | 6.6.300125.1 | $C_6$ (as 6T1) | $0$ | $1$ | |
| 1.35.6t1.b.b | $1$ | $ 5 \cdot 7 $ | 6.6.300125.1 | $C_6$ (as 6T1) | $0$ | $1$ | |
| 1.28.6t1.a.b | $1$ | $ 2^{2} \cdot 7 $ | 6.0.153664.1 | $C_6$ (as 6T1) | $0$ | $-1$ | |
| 1.7.3t1.a.b | $1$ | $ 7 $ | \(\Q(\zeta_{7})^+\) | $C_3$ (as 3T1) | $0$ | $1$ | |
| * | 1.5.4t1.a.a | $1$ | $ 5 $ | \(\Q(\zeta_{5})\) | $C_4$ (as 4T1) | $0$ | $-1$ |
| * | 1.20.4t1.a.a | $1$ | $ 2^{2} \cdot 5 $ | \(\Q(\zeta_{20})^+\) | $C_4$ (as 4T1) | $0$ | $1$ |
| * | 1.20.4t1.a.b | $1$ | $ 2^{2} \cdot 5 $ | \(\Q(\zeta_{20})^+\) | $C_4$ (as 4T1) | $0$ | $1$ |
| * | 1.5.4t1.a.b | $1$ | $ 5 $ | \(\Q(\zeta_{5})\) | $C_4$ (as 4T1) | $0$ | $-1$ |
| 1.140.12t1.a.a | $1$ | $ 2^{2} \cdot 5 \cdot 7 $ | 12.12.46118408000000000.1 | $C_{12}$ (as 12T1) | $0$ | $1$ | |
| 1.35.12t1.a.a | $1$ | $ 5 \cdot 7 $ | 12.0.11259376953125.1 | $C_{12}$ (as 12T1) | $0$ | $-1$ | |
| 1.140.12t1.a.b | $1$ | $ 2^{2} \cdot 5 \cdot 7 $ | 12.12.46118408000000000.1 | $C_{12}$ (as 12T1) | $0$ | $1$ | |
| 1.35.12t1.a.b | $1$ | $ 5 \cdot 7 $ | 12.0.11259376953125.1 | $C_{12}$ (as 12T1) | $0$ | $-1$ | |
| 1.35.12t1.a.c | $1$ | $ 5 \cdot 7 $ | 12.0.11259376953125.1 | $C_{12}$ (as 12T1) | $0$ | $-1$ | |
| 1.35.12t1.a.d | $1$ | $ 5 \cdot 7 $ | 12.0.11259376953125.1 | $C_{12}$ (as 12T1) | $0$ | $-1$ | |
| 1.140.12t1.a.c | $1$ | $ 2^{2} \cdot 5 \cdot 7 $ | 12.12.46118408000000000.1 | $C_{12}$ (as 12T1) | $0$ | $1$ | |
| 1.140.12t1.a.d | $1$ | $ 2^{2} \cdot 5 \cdot 7 $ | 12.12.46118408000000000.1 | $C_{12}$ (as 12T1) | $0$ | $1$ | |
| 2.980.3t2.a.a | $2$ | $ 2^{2} \cdot 5 \cdot 7^{2}$ | 3.1.980.1 | $S_3$ (as 3T2) | $1$ | $0$ | |
| 2.980.6t3.d.a | $2$ | $ 2^{2} \cdot 5 \cdot 7^{2}$ | 6.2.4802000.1 | $D_{6}$ (as 6T3) | $1$ | $0$ | |
| * | 2.140.12t18.a.a | $2$ | $ 2^{2} \cdot 5 \cdot 7 $ | 12.0.153664000000.1 | $C_6\times S_3$ (as 12T18) | $0$ | $0$ |
| * | 2.140.6t5.a.a | $2$ | $ 2^{2} \cdot 5 \cdot 7 $ | 6.0.392000.1 | $S_3\times C_3$ (as 6T5) | $0$ | $0$ |
| * | 2.140.12t18.a.b | $2$ | $ 2^{2} \cdot 5 \cdot 7 $ | 12.0.153664000000.1 | $C_6\times S_3$ (as 12T18) | $0$ | $0$ |
| * | 2.140.6t5.a.b | $2$ | $ 2^{2} \cdot 5 \cdot 7 $ | 6.0.392000.1 | $S_3\times C_3$ (as 6T5) | $0$ | $0$ |
| 2.4900.12t11.a.a | $2$ | $ 2^{2} \cdot 5^{2} \cdot 7^{2}$ | 12.4.46118408000000000.2 | $S_3 \times C_4$ (as 12T11) | $0$ | $0$ | |
| 2.4900.12t11.a.b | $2$ | $ 2^{2} \cdot 5^{2} \cdot 7^{2}$ | 12.4.46118408000000000.2 | $S_3 \times C_4$ (as 12T11) | $0$ | $0$ | |
| * | 2.700.24t65.a.a | $2$ | $ 2^{2} \cdot 5^{2} \cdot 7 $ | 24.0.368947264000000000000000000.1 | $C_{12}\times S_3$ (as 24T65) | $0$ | $0$ |
| * | 2.700.24t65.a.b | $2$ | $ 2^{2} \cdot 5^{2} \cdot 7 $ | 24.0.368947264000000000000000000.1 | $C_{12}\times S_3$ (as 24T65) | $0$ | $0$ |
| * | 2.700.24t65.a.c | $2$ | $ 2^{2} \cdot 5^{2} \cdot 7 $ | 24.0.368947264000000000000000000.1 | $C_{12}\times S_3$ (as 24T65) | $0$ | $0$ |
| * | 2.700.24t65.a.d | $2$ | $ 2^{2} \cdot 5^{2} \cdot 7 $ | 24.0.368947264000000000000000000.1 | $C_{12}\times S_3$ (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 *.