Normalized defining polynomial
\( x^{16} + 780 x^{14} + 188370 x^{12} + 19164600 x^{10} + 930852000 x^{8} + 22422582000 x^{6} + 257859693000 x^{4} + 1249258140000 x^{2} + 1873887210000 \)
Invariants
| Degree: | $16$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[0, 8]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(656523274126409603991207936000000000000=2^{44}\cdot 3^{8}\cdot 5^{12}\cdot 13^{12}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $266.73$ | 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: | $2, 3, 5, 13$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(3120=2^{4}\cdot 3\cdot 5\cdot 13\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{3120}(961,·)$, $\chi_{3120}(1,·)$, $\chi_{3120}(649,·)$, $\chi_{3120}(1249,·)$, $\chi_{3120}(1997,·)$, $\chi_{3120}(1877,·)$, $\chi_{3120}(1561,·)$, $\chi_{3120}(2521,·)$, $\chi_{3120}(1373,·)$, $\chi_{3120}(2209,·)$, $\chi_{3120}(1253,·)$, $\chi_{3120}(317,·)$, $\chi_{3120}(437,·)$, $\chi_{3120}(2809,·)$, $\chi_{3120}(2813,·)$, $\chi_{3120}(2933,·)$$\rbrace$ | ||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $\frac{1}{3} a^{2}$, $\frac{1}{3} a^{3}$, $\frac{1}{1170} a^{4}$, $\frac{1}{1170} a^{5}$, $\frac{1}{10530} a^{6} - \frac{1}{3510} a^{4} + \frac{1}{9} a^{2} - \frac{1}{3}$, $\frac{1}{10530} a^{7} - \frac{1}{3510} a^{5} + \frac{1}{9} a^{3} - \frac{1}{3} a$, $\frac{1}{4106700} a^{8} - \frac{1}{3}$, $\frac{1}{4106700} a^{9} - \frac{1}{3} a$, $\frac{1}{12320100} a^{10} - \frac{1}{9} a^{2}$, $\frac{1}{12320100} a^{11} - \frac{1}{9} a^{3}$, $\frac{1}{1153161360000} a^{12} + \frac{11}{369603000} a^{10} - \frac{1}{8213400} a^{8} - \frac{23}{1263600} a^{6} - \frac{1}{7020} a^{4} - \frac{19}{270} a^{2} - \frac{359}{720}$, $\frac{1}{1153161360000} a^{13} + \frac{11}{369603000} a^{11} - \frac{1}{8213400} a^{9} - \frac{23}{1263600} a^{7} - \frac{1}{7020} a^{5} - \frac{19}{270} a^{3} - \frac{359}{720} a$, $\frac{1}{10378452240000} a^{14} - \frac{7}{221761800} a^{10} - \frac{11}{147841200} a^{8} + \frac{1}{189540} a^{6} - \frac{1}{31590} a^{4} + \frac{227}{2160} a^{2} + \frac{13}{27}$, $\frac{1}{10378452240000} a^{15} - \frac{7}{221761800} a^{11} - \frac{11}{147841200} a^{9} + \frac{1}{189540} a^{7} - \frac{1}{31590} a^{5} + \frac{227}{2160} a^{3} + \frac{13}{27} a$
Class group and class number
$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{4}\times C_{4}\times C_{20}\times C_{40}\times C_{680}$, which has order $139264000$ (assuming GRH)
Unit group
| Rank: | $7$ | 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: | \( 57386.413931756586 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
$C_2^2\times C_4$ (as 16T2):
| An abelian group of order 16 |
| The 16 conjugacy class representatives for $C_4\times C_2^2$ |
| Character table for $C_4\times C_2^2$ |
Intermediate fields
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 | R | ${\href{/LocalNumberField/7.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ | R | ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$ |
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 |
|---|---|---|---|---|---|---|---|
| $2$ | 2.8.22.2 | $x^{8} + 10 x^{4} + 16 x + 4$ | $4$ | $2$ | $22$ | $C_4\times C_2$ | $[3, 4]^{2}$ |
| 2.8.22.2 | $x^{8} + 10 x^{4} + 16 x + 4$ | $4$ | $2$ | $22$ | $C_4\times C_2$ | $[3, 4]^{2}$ | |
| $3$ | 3.4.2.1 | $x^{4} + 9 x^{2} + 36$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
| 3.4.2.1 | $x^{4} + 9 x^{2} + 36$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 3.4.2.1 | $x^{4} + 9 x^{2} + 36$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 3.4.2.1 | $x^{4} + 9 x^{2} + 36$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| $5$ | 5.8.6.1 | $x^{8} - 5 x^{4} + 400$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ |
| 5.8.6.1 | $x^{8} - 5 x^{4} + 400$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ | |
| $13$ | 13.8.6.1 | $x^{8} - 13 x^{4} + 2704$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ |
| 13.8.6.1 | $x^{8} - 13 x^{4} + 2704$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ |