Normalized defining polynomial
\( x^{12} + 1047 x^{10} + 298395 x^{8} + 30228984 x^{6} + 840352563 x^{4} + 783107838 x^{2} + 185472909 \)
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: | \(27936189811629258922498551985852416=2^{12}\cdot 3^{6}\cdot 349^{11}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $742.19$ | 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: | $2, 3, 349$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(4188=2^{2}\cdot 3\cdot 349\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{4188}(1,·)$, $\chi_{4188}(3815,·)$, $\chi_{4188}(2603,·)$, $\chi_{4188}(3961,·)$, $\chi_{4188}(911,·)$, $\chi_{4188}(3863,·)$, $\chi_{4188}(2579,·)$, $\chi_{4188}(887,·)$, $\chi_{4188}(697,·)$, $\chi_{4188}(1273,·)$, $\chi_{4188}(3613,·)$, $\chi_{4188}(925,·)$$\rbrace$ | ||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $\frac{1}{3} a^{2}$, $\frac{1}{3} a^{3}$, $\frac{1}{9} a^{4}$, $\frac{1}{9} a^{5}$, $\frac{1}{27} a^{6}$, $\frac{1}{81} a^{7} + \frac{1}{27} a^{5} - \frac{1}{9} a^{3} + \frac{1}{3} a$, $\frac{1}{37179} a^{8} + \frac{223}{12393} a^{6} + \frac{89}{4131} a^{4} + \frac{55}{1377} a^{2} + \frac{25}{51}$, $\frac{1}{111537} a^{9} + \frac{223}{37179} a^{7} - \frac{370}{12393} a^{5} + \frac{55}{4131} a^{3} + \frac{76}{153} a$, $\frac{1}{17676782179421571} a^{10} + \frac{48445668610}{5892260726473857} a^{8} + \frac{9709297974026}{1964086908824619} a^{6} - \frac{2127837865094}{654695636274873} a^{4} - \frac{2196158151313}{24247986528699} a^{2} + \frac{298958686747}{898073575137}$, $\frac{1}{53030346538264713} a^{11} + \frac{48445668610}{17676782179421571} a^{9} + \frac{9709297974026}{5892260726473857} a^{7} + \frac{70616121721003}{1964086908824619} a^{5} - \frac{10278820327546}{72743959586097} a^{3} - \frac{599114888390}{2694220725411} a$
Class group and class number
$C_{2}\times C_{2}\times C_{4}\times C_{4}\times C_{4}\times C_{4}\times C_{31076}$, which has order $31821824$ (assuming GRH)
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 (assuming GRH) | magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
| |
| Regulator: | \( 49949.033153968034 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A cyclic group of order 12 |
| The 12 conjugacy class representatives for $C_{12}$ |
| Character table for $C_{12}$ |
Intermediate fields
| \(\Q(\sqrt{349}) \), 3.3.121801.1, 4.0.6121231056.1, 6.6.5177583776749.1 |
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 | R | R | ${\href{/LocalNumberField/5.3.0.1}{3} }^{4}$ | ${\href{/LocalNumberField/7.12.0.1}{12} }$ | ${\href{/LocalNumberField/11.4.0.1}{4} }^{3}$ | ${\href{/LocalNumberField/13.12.0.1}{12} }$ | ${\href{/LocalNumberField/17.1.0.1}{1} }^{12}$ | ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}$ | ${\href{/LocalNumberField/23.3.0.1}{3} }^{4}$ | ${\href{/LocalNumberField/29.3.0.1}{3} }^{4}$ | ${\href{/LocalNumberField/31.2.0.1}{2} }^{6}$ | ${\href{/LocalNumberField/37.2.0.1}{2} }^{6}$ | ${\href{/LocalNumberField/41.2.0.1}{2} }^{6}$ | ${\href{/LocalNumberField/43.12.0.1}{12} }$ | ${\href{/LocalNumberField/47.4.0.1}{4} }^{3}$ | ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}$ | ${\href{/LocalNumberField/59.12.0.1}{12} }$ |
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 |
|---|---|---|---|---|---|---|---|
| $2$ | 2.12.12.25 | $x^{12} - 78 x^{10} - 1621 x^{8} + 460 x^{6} - 1977 x^{4} + 866 x^{2} + 749$ | $2$ | $6$ | $12$ | $C_{12}$ | $[2]^{6}$ |
| $3$ | 3.6.3.2 | $x^{6} - 9 x^{2} + 27$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ |
| 3.6.3.2 | $x^{6} - 9 x^{2} + 27$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
| 349 | Data not computed | ||||||