Normalized defining polynomial
\( x^{12} - x^{11} - 81 x^{10} - 55 x^{9} + 2187 x^{8} + 4623 x^{7} - 19442 x^{6} - 69890 x^{5} + 4263 x^{4} + 257547 x^{3} + 289908 x^{2} - 19988 x - 119128 \)
Invariants
| Degree: | $12$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[12, 0]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(4891005035897482905857=257^{9}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $64.19$ | magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
| |
| Ramified primes: | $257$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois over $\Q$. | |||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{6} a^{8} - \frac{1}{6} a^{7} - \frac{1}{6} a^{6} - \frac{1}{6} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{12} a^{9} - \frac{1}{12} a^{8} + \frac{1}{6} a^{7} - \frac{1}{12} a^{5} - \frac{1}{12} a^{4} - \frac{5}{12} a^{3} + \frac{5}{12} a^{2} - \frac{1}{6} a$, $\frac{1}{12} a^{10} - \frac{1}{12} a^{8} - \frac{1}{6} a^{7} + \frac{1}{12} a^{6} - \frac{1}{6} a^{5} + \frac{1}{6} a^{4} + \frac{1}{6} a^{3} + \frac{1}{12} a^{2} - \frac{1}{2} a + \frac{1}{3}$, $\frac{1}{365276900168592} a^{11} + \frac{5797912272233}{182638450084296} a^{10} + \frac{1083729290491}{121758966722864} a^{9} - \frac{1929969629467}{91319225042148} a^{8} + \frac{7966124650315}{365276900168592} a^{7} + \frac{1985976044827}{45659612521074} a^{6} - \frac{147036897071}{946313212872} a^{5} - \frac{15184704328695}{30439741680716} a^{4} + \frac{54801618786499}{365276900168592} a^{3} + \frac{14931018576893}{30439741680716} a^{2} + \frac{36060242789615}{91319225042148} a - \frac{17053755978565}{45659612521074}$
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
| Rank: | $11$ | magma: UnitRank(K);
sage: UK.rank()
gp: K.fu
| |
| Torsion generator: | \( -1 \) (order $2$) | magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
sage: UK.torsion_generator()
gp: K.tu[2]
| |
| 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) | magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
| |
| Regulator: | \( 9021039.03079 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A solvable group of order 12 |
| The 6 conjugacy class representatives for $C_3 : C_4$ |
| Character table for $C_3 : C_4$ |
Intermediate fields
| \(\Q(\sqrt{257}) \), 3.3.257.1 x3, 4.4.16974593.1, 6.6.16974593.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Frobenius cycle types
| $p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 | 53 | 59 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/LocalNumberField/2.3.0.1}{3} }^{4}$ | ${\href{/LocalNumberField/3.4.0.1}{4} }^{3}$ | ${\href{/LocalNumberField/5.4.0.1}{4} }^{3}$ | ${\href{/LocalNumberField/7.4.0.1}{4} }^{3}$ | ${\href{/LocalNumberField/11.3.0.1}{3} }^{4}$ | ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}$ | ${\href{/LocalNumberField/17.3.0.1}{3} }^{4}$ | ${\href{/LocalNumberField/19.4.0.1}{4} }^{3}$ | ${\href{/LocalNumberField/23.3.0.1}{3} }^{4}$ | ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}$ | ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}$ | ${\href{/LocalNumberField/37.4.0.1}{4} }^{3}$ | ${\href{/LocalNumberField/41.4.0.1}{4} }^{3}$ | ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}$ | ${\href{/LocalNumberField/47.4.0.1}{4} }^{3}$ | ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}$ | ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}$ |
Cycle lengths which are repeated in a cycle type are indicated by exponents.
Local algebras for ramified primes
| $p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
|---|---|---|---|---|---|---|---|
| 257 | Data not computed | ||||||
Artin representations
| Label | Dimension | Conductor | Defining polynomial of Artin field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| * | 1.1.1t1.1c1 | $1$ | $1$ | $x$ | $C_1$ | $1$ | $1$ |
| * | 1.257.2t1.1c1 | $1$ | $ 257 $ | $x^{2} - x - 64$ | $C_2$ (as 2T1) | $1$ | $1$ |
| * | 1.257.4t1.1c1 | $1$ | $ 257 $ | $x^{4} - x^{3} - 96 x^{2} + 16 x + 256$ | $C_4$ (as 4T1) | $0$ | $1$ |
| * | 1.257.4t1.1c2 | $1$ | $ 257 $ | $x^{4} - x^{3} - 96 x^{2} + 16 x + 256$ | $C_4$ (as 4T1) | $0$ | $1$ |
| *2 | 2.257e2.12t5.1c1 | $2$ | $ 257^{2}$ | $x^{12} - x^{11} - 81 x^{10} - 55 x^{9} + 2187 x^{8} + 4623 x^{7} - 19442 x^{6} - 69890 x^{5} + 4263 x^{4} + 257547 x^{3} + 289908 x^{2} - 19988 x - 119128$ | $C_3 : C_4$ (as 12T5) | $-1$ | $2$ |
| *2 | 2.257.3t2.1c1 | $2$ | $ 257 $ | $x^{3} - x^{2} - 4 x + 3$ | $S_3$ (as 3T2) | $1$ | $2$ |