Normalized defining polynomial
\( x^{12} - x^{11} + 7 x^{10} - 14 x^{9} + 62 x^{8} - 99 x^{7} + 248 x^{6} - 282 x^{5} + 513 x^{4} - 323 x^{3} + 547 x^{2} - 243 x + 239 \)
Invariants
| Degree: | $12$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[0, 6]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(61243533037578125=5^{7}\cdot 13^{8}\cdot 31^{2}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $25.06$ | 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: | $5, 13, 31$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is not Galois over $\Q$. | |||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{55} a^{10} + \frac{1}{11} a^{9} - \frac{27}{55} a^{8} - \frac{1}{55} a^{7} + \frac{24}{55} a^{6} - \frac{1}{55} a^{5} + \frac{26}{55} a^{4} - \frac{7}{55} a^{3} + \frac{17}{55} a^{2} + \frac{7}{55} a - \frac{4}{55}$, $\frac{1}{153281925775} a^{11} - \frac{433708988}{153281925775} a^{10} + \frac{45139191088}{153281925775} a^{9} + \frac{4778019911}{30656385155} a^{8} - \frac{58112515973}{153281925775} a^{7} - \frac{69825805298}{153281925775} a^{6} - \frac{16357992226}{153281925775} a^{5} - \frac{3757931854}{30656385155} a^{4} - \frac{72213893472}{153281925775} a^{3} - \frac{3291356394}{13934720525} a^{2} + \frac{7885222406}{30656385155} a - \frac{71511114553}{153281925775}$
Class group and class number
$C_{2}\times C_{2}$, which has order $4$
Unit group
| Rank: | $5$ | 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 | magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
| |
| Regulator: | \( 615.54450504 \) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
$C_2^5.C_6$ (as 12T105):
| A solvable group of order 192 |
| The 20 conjugacy class representatives for $C_2^5.C_6$ |
| Character table for $C_2^5.C_6$ |
Intermediate fields
| \(\Q(\sqrt{5}) \), 3.3.169.1, 6.6.3570125.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Degree 12 siblings: | data not computed |
| Degree 16 siblings: | data not computed |
| Degree 24 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 | ${\href{/LocalNumberField/2.12.0.1}{12} }$ | ${\href{/LocalNumberField/3.12.0.1}{12} }$ | R | ${\href{/LocalNumberField/7.12.0.1}{12} }$ | ${\href{/LocalNumberField/11.3.0.1}{3} }^{4}$ | R | ${\href{/LocalNumberField/17.12.0.1}{12} }$ | ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}$ | ${\href{/LocalNumberField/23.12.0.1}{12} }$ | ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}$ | R | ${\href{/LocalNumberField/37.12.0.1}{12} }$ | ${\href{/LocalNumberField/41.3.0.1}{3} }^{4}$ | ${\href{/LocalNumberField/43.12.0.1}{12} }$ | ${\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}$ |
In the table, R denotes a ramified prime. 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 |
|---|---|---|---|---|---|---|---|
| $5$ | 5.2.1.1 | $x^{2} - 5$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 5.2.1.1 | $x^{2} - 5$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 5.4.2.1 | $x^{4} + 15 x^{2} + 100$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 5.4.3.1 | $x^{4} - 5$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ | |
| $13$ | 13.12.8.1 | $x^{12} - 39 x^{9} - 338 x^{6} + 10985 x^{3} + 228488$ | $3$ | $4$ | $8$ | $C_{12}$ | $[\ ]_{3}^{4}$ |
| $31$ | $\Q_{31}$ | $x + 7$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{31}$ | $x + 7$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{31}$ | $x + 7$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{31}$ | $x + 7$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 31.2.1.2 | $x^{2} + 217$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 31.2.1.2 | $x^{2} + 217$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 31.2.0.1 | $x^{2} - x + 12$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 31.2.0.1 | $x^{2} - x + 12$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |