Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^24 - 3*x^16 - 8*x^12 + 18*x^8 + 8*x^4 + 1)
gp: K = bnfinit(x^24 - 3*x^16 - 8*x^12 + 18*x^8 + 8*x^4 + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 0, 0, 0, 8, 0, 0, 0, 18, 0, 0, 0, -8, 0, 0, 0, -3, 0, 0, 0, 0, 0, 0, 0, 1]);
\( x^{24} - 3 x^{16} - 8 x^{12} + 18 x^{8} + 8 x^{4} + 1 \)
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: | \(3852179415897489839182437154816\)\(\medspace = 2^{72}\cdot 13^{8}\) | sage: K.disc()
gp: K.disc
magma: Discriminant(Integers(K));
| |
| Root discriminant: | $18.81$ | 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: | $2, 13$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $\frac{1}{899} a^{20} + \frac{287}{899} a^{16} - \frac{342}{899} a^{12} - \frac{171}{899} a^{8} + \frac{386}{899} a^{4} + \frac{213}{899}$, $\frac{1}{899} a^{21} + \frac{287}{899} a^{17} - \frac{342}{899} a^{13} - \frac{171}{899} a^{9} + \frac{386}{899} a^{5} + \frac{213}{899} a$, $\frac{1}{899} a^{22} + \frac{287}{899} a^{18} - \frac{342}{899} a^{14} - \frac{171}{899} a^{10} + \frac{386}{899} a^{6} + \frac{213}{899} a^{2}$, $\frac{1}{899} a^{23} + \frac{287}{899} a^{19} - \frac{342}{899} a^{15} - \frac{171}{899} a^{11} + \frac{386}{899} a^{7} + \frac{213}{899} a^{3}$
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: | \( \frac{952}{899} a^{23} - \frac{72}{899} a^{19} - \frac{2843}{899} a^{15} - \frac{7265}{899} a^{11} + \frac{17761}{899} a^{7} + \frac{5895}{899} a^{3} \) (order $16$) | 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: | \( 2446825.407448486 \) (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
| \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-2}) \), \(\Q(\sqrt{-1}) \), 4.0.2048.2, \(\Q(\zeta_{16})^+\), \(\Q(\zeta_{8})\), 6.0.10816.1, \(\Q(\zeta_{16})\), 12.0.479174066176.4 |
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 | ${\href{/LocalNumberField/3.12.0.1}{12} }^{2}$ | ${\href{/LocalNumberField/5.12.0.1}{12} }^{2}$ | ${\href{/LocalNumberField/7.6.0.1}{6} }^{4}$ | ${\href{/LocalNumberField/11.12.0.1}{12} }^{2}$ | R | ${\href{/LocalNumberField/17.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{12}$ | ${\href{/LocalNumberField/19.12.0.1}{12} }^{2}$ | ${\href{/LocalNumberField/23.6.0.1}{6} }^{4}$ | ${\href{/LocalNumberField/29.12.0.1}{12} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }^{3}$ | ${\href{/LocalNumberField/31.2.0.1}{2} }^{12}$ | ${\href{/LocalNumberField/37.12.0.1}{12} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }^{3}$ | ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{6}$ | ${\href{/LocalNumberField/43.12.0.1}{12} }^{2}$ | ${\href{/LocalNumberField/47.2.0.1}{2} }^{12}$ | ${\href{/LocalNumberField/53.12.0.1}{12} }^{2}$ | ${\href{/LocalNumberField/59.12.0.1}{12} }^{2}$ |
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
| $p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
|---|---|---|---|---|---|---|---|
| 2 | Data not computed | ||||||
| $13$ | 13.12.8.2 | $x^{12} + 169 x^{6} - 2197 x^{3} + 57122$ | $3$ | $4$ | $8$ | $C_{12}$ | $[\ ]_{3}^{4}$ |
| 13.12.0.1 | $x^{12} + x^{2} - x + 2$ | $1$ | $12$ | $0$ | $C_{12}$ | $[\ ]^{12}$ | |
Artin representations
| Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| * | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
| * | 1.8.2t1.b.a | $1$ | $ 2^{3}$ | \(\Q(\sqrt{-2}) \) | $C_2$ (as 2T1) | $1$ | $-1$ |
| * | 1.4.2t1.a.a | $1$ | $ 2^{2}$ | \(\Q(\sqrt{-1}) \) | $C_2$ (as 2T1) | $1$ | $-1$ |
| * | 1.8.2t1.a.a | $1$ | $ 2^{3}$ | \(\Q(\sqrt{2}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
| 1.52.6t1.b.a | $1$ | $ 2^{2} \cdot 13 $ | 6.0.1827904.1 | $C_6$ (as 6T1) | $0$ | $-1$ | |
| 1.52.6t1.b.b | $1$ | $ 2^{2} \cdot 13 $ | 6.0.1827904.1 | $C_6$ (as 6T1) | $0$ | $-1$ | |
| 1.104.6t1.d.a | $1$ | $ 2^{3} \cdot 13 $ | 6.0.14623232.1 | $C_6$ (as 6T1) | $0$ | $-1$ | |
| 1.13.3t1.a.a | $1$ | $ 13 $ | 3.3.169.1 | $C_3$ (as 3T1) | $0$ | $1$ | |
| 1.13.3t1.a.b | $1$ | $ 13 $ | 3.3.169.1 | $C_3$ (as 3T1) | $0$ | $1$ | |
| 1.104.6t1.a.a | $1$ | $ 2^{3} \cdot 13 $ | 6.6.14623232.1 | $C_6$ (as 6T1) | $0$ | $1$ | |
| 1.104.6t1.a.b | $1$ | $ 2^{3} \cdot 13 $ | 6.6.14623232.1 | $C_6$ (as 6T1) | $0$ | $1$ | |
| 1.104.6t1.d.b | $1$ | $ 2^{3} \cdot 13 $ | 6.0.14623232.1 | $C_6$ (as 6T1) | $0$ | $-1$ | |
| * | 1.16.4t1.a.a | $1$ | $ 2^{4}$ | \(\Q(\zeta_{16})^+\) | $C_4$ (as 4T1) | $0$ | $1$ |
| * | 1.16.4t1.b.a | $1$ | $ 2^{4}$ | 4.0.2048.2 | $C_4$ (as 4T1) | $0$ | $-1$ |
| * | 1.16.4t1.b.b | $1$ | $ 2^{4}$ | 4.0.2048.2 | $C_4$ (as 4T1) | $0$ | $-1$ |
| * | 1.16.4t1.a.b | $1$ | $ 2^{4}$ | \(\Q(\zeta_{16})^+\) | $C_4$ (as 4T1) | $0$ | $1$ |
| 1.208.12t1.b.a | $1$ | $ 2^{4} \cdot 13 $ | 12.12.7007073538075000832.1 | $C_{12}$ (as 12T1) | $0$ | $1$ | |
| 1.208.12t1.a.a | $1$ | $ 2^{4} \cdot 13 $ | 12.0.7007073538075000832.1 | $C_{12}$ (as 12T1) | $0$ | $-1$ | |
| 1.208.12t1.a.b | $1$ | $ 2^{4} \cdot 13 $ | 12.0.7007073538075000832.1 | $C_{12}$ (as 12T1) | $0$ | $-1$ | |
| 1.208.12t1.b.b | $1$ | $ 2^{4} \cdot 13 $ | 12.12.7007073538075000832.1 | $C_{12}$ (as 12T1) | $0$ | $1$ | |
| 1.208.12t1.b.c | $1$ | $ 2^{4} \cdot 13 $ | 12.12.7007073538075000832.1 | $C_{12}$ (as 12T1) | $0$ | $1$ | |
| 1.208.12t1.b.d | $1$ | $ 2^{4} \cdot 13 $ | 12.12.7007073538075000832.1 | $C_{12}$ (as 12T1) | $0$ | $1$ | |
| 1.208.12t1.a.c | $1$ | $ 2^{4} \cdot 13 $ | 12.0.7007073538075000832.1 | $C_{12}$ (as 12T1) | $0$ | $-1$ | |
| 1.208.12t1.a.d | $1$ | $ 2^{4} \cdot 13 $ | 12.0.7007073538075000832.1 | $C_{12}$ (as 12T1) | $0$ | $-1$ | |
| 2.676.3t2.b.a | $2$ | $ 2^{2} \cdot 13^{2}$ | 3.1.676.1 | $S_3$ (as 3T2) | $1$ | $0$ | |
| 2.10816.6t3.d.a | $2$ | $ 2^{6} \cdot 13^{2}$ | 6.0.58492928.3 | $D_{6}$ (as 6T3) | $1$ | $0$ | |
| * | 2.832.12t18.b.a | $2$ | $ 2^{6} \cdot 13 $ | 12.0.479174066176.4 | $C_6\times S_3$ (as 12T18) | $0$ | $0$ |
| * | 2.832.12t18.b.b | $2$ | $ 2^{6} \cdot 13 $ | 12.0.479174066176.4 | $C_6\times S_3$ (as 12T18) | $0$ | $0$ |
| * | 2.52.6t5.b.a | $2$ | $ 2^{2} \cdot 13 $ | 6.0.10816.1 | $S_3\times C_3$ (as 6T5) | $0$ | $0$ |
| * | 2.52.6t5.b.b | $2$ | $ 2^{2} \cdot 13 $ | 6.0.10816.1 | $S_3\times C_3$ (as 6T5) | $0$ | $0$ |
| 2.43264.12t11.b.a | $2$ | $ 2^{8} \cdot 13^{2}$ | 12.4.28028294152300003328.26 | $S_3 \times C_4$ (as 12T11) | $0$ | $0$ | |
| 2.43264.12t11.b.b | $2$ | $ 2^{8} \cdot 13^{2}$ | 12.4.28028294152300003328.26 | $S_3 \times C_4$ (as 12T11) | $0$ | $0$ | |
| * | 2.3328.24t65.a.a | $2$ | $ 2^{8} \cdot 13 $ | 24.0.3852179415897489839182437154816.1 | $C_{12}\times S_3$ (as 24T65) | $0$ | $0$ |
| * | 2.3328.24t65.a.b | $2$ | $ 2^{8} \cdot 13 $ | 24.0.3852179415897489839182437154816.1 | $C_{12}\times S_3$ (as 24T65) | $0$ | $0$ |
| * | 2.3328.24t65.a.c | $2$ | $ 2^{8} \cdot 13 $ | 24.0.3852179415897489839182437154816.1 | $C_{12}\times S_3$ (as 24T65) | $0$ | $0$ |
| * | 2.3328.24t65.a.d | $2$ | $ 2^{8} \cdot 13 $ | 24.0.3852179415897489839182437154816.1 | $C_{12}\times S_3$ (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 *.