Normalized defining polynomial
\( x^{12} + 1422 x^{8} - 924300 x^{6} - 11027847 x^{4} + 657177300 x^{2} + 225275535424 \)
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: | \(1669440466451705194787135164416=2^{12}\cdot 3^{16}\cdot 79^{10}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $330.03$ | 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: | $2, 3, 79$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(2844=2^{2}\cdot 3^{2}\cdot 79\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{2844}(1,·)$, $\chi_{2844}(1003,·)$, $\chi_{2844}(2053,·)$, $\chi_{2844}(103,·)$, $\chi_{2844}(925,·)$, $\chi_{2844}(2347,·)$, $\chi_{2844}(655,·)$, $\chi_{2844}(1525,·)$, $\chi_{2844}(631,·)$, $\chi_{2844}(2425,·)$, $\chi_{2844}(1423,·)$, $\chi_{2844}(2077,·)$$\rbrace$ | ||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $\frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{20} a^{4} + \frac{1}{4} a^{2} + \frac{1}{5}$, $\frac{1}{40} a^{5} + \frac{1}{8} a^{3} + \frac{1}{10} a$, $\frac{1}{3160} a^{6} - \frac{1}{40} a^{4} + \frac{2}{5} a^{2} + \frac{2}{5}$, $\frac{1}{6320} a^{7} - \frac{19}{80} a^{3} - \frac{1}{4} a$, $\frac{1}{1264000} a^{8} - \frac{13}{126400} a^{6} - \frac{353}{16000} a^{4} - \frac{129}{800} a^{2} - \frac{69}{250}$, $\frac{1}{949264000} a^{9} + \frac{6707}{94926400} a^{7} + \frac{103647}{12016000} a^{5} - \frac{38369}{600800} a^{3} + \frac{43903}{93875} a$, $\frac{1}{4313455616000} a^{10} + \frac{1063647}{4313455616000} a^{8} - \frac{636853297}{4313455616000} a^{6} - \frac{1027858261}{54600704000} a^{4} - \frac{6264574869}{13650176000} a^{2} + \frac{376737}{1136000}$, $\frac{1}{340762993664000} a^{11} + \frac{177}{4313455616000} a^{9} + \frac{102951937}{4313455616000} a^{7} + \frac{37599517931}{4313455616000} a^{5} - \frac{2463345691}{13650176000} a^{3} + \frac{298043457}{853136000} a$
Class group and class number
$C_{2}\times C_{2}\times C_{4}\times C_{4}\times C_{168}\times C_{2520}$, which has order $27095040$ (assuming GRH)
Unit group
| Rank: | $5$ | magma: UnitRank(K);
sage: UK.rank()
gp: K.fu
| |
| Torsion generator: | \( -\frac{9}{149983712000} a^{11} - \frac{1}{1898528000} a^{9} - \frac{393}{1898528000} a^{7} + \frac{50517}{1898528000} a^{5} + \frac{4879}{6008000} a^{3} + \frac{207}{187750} a \) (order $4$) | 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: | \( 447819.48619075713 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
$C_2\times C_6$ (as 12T2):
| An abelian group of order 12 |
| The 12 conjugacy class representatives for $C_6\times C_2$ |
| Character table for $C_6\times C_2$ |
Intermediate fields
| \(\Q(\sqrt{-1}) \), \(\Q(\sqrt{79}) \), \(\Q(\sqrt{-79}) \), 3.3.505521.1, \(\Q(i, \sqrt{79})\), 6.0.16355294812224.1, 6.6.1292068290165696.1, 6.0.20188567033839.2 |
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 | ${\href{/LocalNumberField/5.1.0.1}{1} }^{12}$ | ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}$ | ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}$ | ${\href{/LocalNumberField/13.3.0.1}{3} }^{4}$ | ${\href{/LocalNumberField/17.2.0.1}{2} }^{6}$ | ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}$ | ${\href{/LocalNumberField/23.2.0.1}{2} }^{6}$ | ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}$ | ${\href{/LocalNumberField/31.2.0.1}{2} }^{6}$ | ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}$ | ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}$ | ${\href{/LocalNumberField/43.2.0.1}{2} }^{6}$ | ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}$ | ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}$ | ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
| $2$ | 2.2.2.1 | $x^{2} + 2 x + 2$ | $2$ | $1$ | $2$ | $C_2$ | $[2]$ |
| 2.2.2.1 | $x^{2} + 2 x + 2$ | $2$ | $1$ | $2$ | $C_2$ | $[2]$ | |
| 2.2.2.1 | $x^{2} + 2 x + 2$ | $2$ | $1$ | $2$ | $C_2$ | $[2]$ | |
| 2.2.2.1 | $x^{2} + 2 x + 2$ | $2$ | $1$ | $2$ | $C_2$ | $[2]$ | |
| 2.2.2.1 | $x^{2} + 2 x + 2$ | $2$ | $1$ | $2$ | $C_2$ | $[2]$ | |
| 2.2.2.1 | $x^{2} + 2 x + 2$ | $2$ | $1$ | $2$ | $C_2$ | $[2]$ | |
| $3$ | 3.6.8.4 | $x^{6} + 18 x^{2} + 63$ | $3$ | $2$ | $8$ | $C_6$ | $[2]^{2}$ |
| 3.6.8.4 | $x^{6} + 18 x^{2} + 63$ | $3$ | $2$ | $8$ | $C_6$ | $[2]^{2}$ | |
| $79$ | 79.12.10.3 | $x^{12} - 553 x^{6} + 505521$ | $6$ | $2$ | $10$ | $C_6\times C_2$ | $[\ ]_{6}^{2}$ |