Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^24 - 2*x^21 - 2*x^15 + 9*x^12 - 8*x^9 + 5*x^6 - 3*x^3 + 1)
gp: K = bnfinit(x^24 - 2*x^21 - 2*x^15 + 9*x^12 - 8*x^9 + 5*x^6 - 3*x^3 + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 0, 0, -3, 0, 0, 5, 0, 0, -8, 0, 0, 9, 0, 0, -2, 0, 0, 0, 0, 0, -2, 0, 0, 1]);
\( x^{24} - 2 x^{21} - 2 x^{15} + 9 x^{12} - 8 x^{9} + 5 x^{6} - 3 x^{3} + 1 \)
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: | \(572565594852444156646728515625\)\(\medspace = 3^{36}\cdot 5^{18}\) | sage: K.disc()
gp: K.disc
magma: Discriminant(Integers(K));
| |
| Root discriminant: | $17.37$ | 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: | $3, 5$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $\frac{1}{61} a^{21} - \frac{26}{61} a^{18} + \frac{14}{61} a^{15} + \frac{28}{61} a^{12} + \frac{8}{61} a^{9} - \frac{17}{61} a^{6} - \frac{14}{61} a^{3} + \frac{28}{61}$, $\frac{1}{61} a^{22} - \frac{26}{61} a^{19} + \frac{14}{61} a^{16} + \frac{28}{61} a^{13} + \frac{8}{61} a^{10} - \frac{17}{61} a^{7} - \frac{14}{61} a^{4} + \frac{28}{61} a$, $\frac{1}{61} a^{23} - \frac{26}{61} a^{20} + \frac{14}{61} a^{17} + \frac{28}{61} a^{14} + \frac{8}{61} a^{11} - \frac{17}{61} a^{8} - \frac{14}{61} a^{5} + \frac{28}{61} a^{2}$
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: | \( -\frac{19}{61} a^{21} + \frac{6}{61} a^{18} + \frac{39}{61} a^{15} + \frac{78}{61} a^{12} - \frac{91}{61} a^{9} - \frac{104}{61} a^{6} - \frac{39}{61} a^{3} + \frac{17}{61} \) (order $30$) | 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: | \( 1414385.0473488418 \) (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
| \(\Q(\sqrt{-15}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-3}) \), \(\Q(\zeta_{15})^+\), \(\Q(\zeta_{5})\), \(\Q(\sqrt{-3}, \sqrt{5})\), 6.0.2460375.2, \(\Q(\zeta_{15})\), 12.0.6053445140625.2 |
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 | ${\href{/LocalNumberField/2.12.0.1}{12} }{,}\,{\href{/LocalNumberField/2.4.0.1}{4} }^{3}$ | R | R | ${\href{/LocalNumberField/7.12.0.1}{12} }^{2}$ | ${\href{/LocalNumberField/11.6.0.1}{6} }^{4}$ | ${\href{/LocalNumberField/13.12.0.1}{12} }^{2}$ | ${\href{/LocalNumberField/17.12.0.1}{12} }^{2}$ | ${\href{/LocalNumberField/19.6.0.1}{6} }^{4}$ | ${\href{/LocalNumberField/23.12.0.1}{12} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }^{3}$ | ${\href{/LocalNumberField/29.6.0.1}{6} }^{4}$ | ${\href{/LocalNumberField/31.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{12}$ | ${\href{/LocalNumberField/37.4.0.1}{4} }^{6}$ | ${\href{/LocalNumberField/41.6.0.1}{6} }^{4}$ | ${\href{/LocalNumberField/43.12.0.1}{12} }^{2}$ | ${\href{/LocalNumberField/47.12.0.1}{12} }^{2}$ | ${\href{/LocalNumberField/53.12.0.1}{12} }^{2}$ | ${\href{/LocalNumberField/59.6.0.1}{6} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
| 3 | Data not computed | ||||||
| 5 | Data 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) | $1$ | $-1$ |
| * | 1.15.2t1.a.a | $1$ | $ 3 \cdot 5 $ | \(\Q(\sqrt{-15}) \) | $C_2$ (as 2T1) | $1$ | $-1$ |
| * | 1.5.2t1.a.a | $1$ | $ 5 $ | \(\Q(\sqrt{5}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
| 1.45.6t1.b.a | $1$ | $ 3^{2} \cdot 5 $ | 6.0.2460375.1 | $C_6$ (as 6T1) | $0$ | $-1$ | |
| 1.45.6t1.a.a | $1$ | $ 3^{2} \cdot 5 $ | 6.6.820125.1 | $C_6$ (as 6T1) | $0$ | $1$ | |
| 1.9.6t1.a.a | $1$ | $ 3^{2}$ | \(\Q(\zeta_{9})\) | $C_6$ (as 6T1) | $0$ | $-1$ | |
| 1.9.3t1.a.a | $1$ | $ 3^{2}$ | \(\Q(\zeta_{9})^+\) | $C_3$ (as 3T1) | $0$ | $1$ | |
| 1.9.6t1.a.b | $1$ | $ 3^{2}$ | \(\Q(\zeta_{9})\) | $C_6$ (as 6T1) | $0$ | $-1$ | |
| 1.9.3t1.a.b | $1$ | $ 3^{2}$ | \(\Q(\zeta_{9})^+\) | $C_3$ (as 3T1) | $0$ | $1$ | |
| 1.45.6t1.b.b | $1$ | $ 3^{2} \cdot 5 $ | 6.0.2460375.1 | $C_6$ (as 6T1) | $0$ | $-1$ | |
| 1.45.6t1.a.b | $1$ | $ 3^{2} \cdot 5 $ | 6.6.820125.1 | $C_6$ (as 6T1) | $0$ | $1$ | |
| * | 1.5.4t1.a.a | $1$ | $ 5 $ | \(\Q(\zeta_{5})\) | $C_4$ (as 4T1) | $0$ | $-1$ |
| * | 1.15.4t1.a.a | $1$ | $ 3 \cdot 5 $ | \(\Q(\zeta_{15})^+\) | $C_4$ (as 4T1) | $0$ | $1$ |
| * | 1.15.4t1.a.b | $1$ | $ 3 \cdot 5 $ | \(\Q(\zeta_{15})^+\) | $C_4$ (as 4T1) | $0$ | $1$ |
| * | 1.5.4t1.a.b | $1$ | $ 5 $ | \(\Q(\zeta_{5})\) | $C_4$ (as 4T1) | $0$ | $-1$ |
| 1.45.12t1.b.a | $1$ | $ 3^{2} \cdot 5 $ | \(\Q(\zeta_{45})^+\) | $C_{12}$ (as 12T1) | $0$ | $1$ | |
| 1.45.12t1.b.b | $1$ | $ 3^{2} \cdot 5 $ | \(\Q(\zeta_{45})^+\) | $C_{12}$ (as 12T1) | $0$ | $1$ | |
| 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) | $0$ | $-1$ | |
| 1.45.12t1.b.c | $1$ | $ 3^{2} \cdot 5 $ | \(\Q(\zeta_{45})^+\) | $C_{12}$ (as 12T1) | $0$ | $1$ | |
| 1.45.12t1.b.d | $1$ | $ 3^{2} \cdot 5 $ | \(\Q(\zeta_{45})^+\) | $C_{12}$ (as 12T1) | $0$ | $1$ | |
| 1.45.12t1.a.c | $1$ | $ 3^{2} \cdot 5 $ | 12.0.84075626953125.1 | $C_{12}$ (as 12T1) | $0$ | $-1$ | |
| 1.45.12t1.a.d | $1$ | $ 3^{2} \cdot 5 $ | 12.0.84075626953125.1 | $C_{12}$ (as 12T1) | $0$ | $-1$ | |
| 2.135.3t2.b.a | $2$ | $ 3^{3} \cdot 5 $ | 3.1.135.1 | $S_3$ (as 3T2) | $1$ | $0$ | |
| 2.135.6t3.a.a | $2$ | $ 3^{3} \cdot 5 $ | 6.2.91125.1 | $D_{6}$ (as 6T3) | $1$ | $0$ | |
| * | 2.405.6t5.a.a | $2$ | $ 3^{4} \cdot 5 $ | 6.0.2460375.2 | $S_3\times C_3$ (as 6T5) | $0$ | $0$ |
| * | 2.405.12t18.a.a | $2$ | $ 3^{4} \cdot 5 $ | 12.0.6053445140625.2 | $C_6\times S_3$ (as 12T18) | $0$ | $0$ |
| * | 2.405.6t5.a.b | $2$ | $ 3^{4} \cdot 5 $ | 6.0.2460375.2 | $S_3\times C_3$ (as 6T5) | $0$ | $0$ |
| * | 2.405.12t18.a.b | $2$ | $ 3^{4} \cdot 5 $ | 12.0.6053445140625.2 | $C_6\times S_3$ (as 12T18) | $0$ | $0$ |
| 2.675.12t11.a.a | $2$ | $ 3^{3} \cdot 5^{2}$ | 12.0.1037970703125.1 | $S_3 \times C_4$ (as 12T11) | $0$ | $0$ | |
| 2.675.12t11.a.b | $2$ | $ 3^{3} \cdot 5^{2}$ | 12.0.1037970703125.1 | $S_3 \times C_4$ (as 12T11) | $0$ | $0$ | |
| * | 2.2025.24t65.a.a | $2$ | $ 3^{4} \cdot 5^{2}$ | 24.0.572565594852444156646728515625.2 | $C_{12}\times S_3$ (as 24T65) | $0$ | $0$ |
| * | 2.2025.24t65.a.b | $2$ | $ 3^{4} \cdot 5^{2}$ | 24.0.572565594852444156646728515625.2 | $C_{12}\times S_3$ (as 24T65) | $0$ | $0$ |
| * | 2.2025.24t65.a.c | $2$ | $ 3^{4} \cdot 5^{2}$ | 24.0.572565594852444156646728515625.2 | $C_{12}\times S_3$ (as 24T65) | $0$ | $0$ |
| * | 2.2025.24t65.a.d | $2$ | $ 3^{4} \cdot 5^{2}$ | 24.0.572565594852444156646728515625.2 | $C_{12}\times S_3$ (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 *.