Normalized defining polynomial
\( x^{16} + 624 x^{14} + 148356 x^{12} + 17010864 x^{10} + 991343691 x^{8} + 27910775880 x^{6} + 314044277040 x^{4} + 936943605000 x^{2} + 566850881025 \)
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: | \(10504372386022553663859326976000000000000=2^{48}\cdot 3^{8}\cdot 5^{12}\cdot 13^{12}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $317.20$ | 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$$\chi_{3120}(1,·)$, $\chi_{3120}(2887,·)$, $\chi_{3120}(649,·)$, $\chi_{3120}(2699,·)$, $\chi_{3120}(1997,·)$, $\chi_{3120}(2579,·)$, $\chi_{3120}(1877,·)$, $\chi_{3120}(2263,·)$, $\chi_{3120}(2521,·)$, $\chi_{3120}(1373,·)$, $\chi_{3120}(1249,·)$, $\chi_{3120}(1253,·)$, $\chi_{3120}(1451,·)$, $\chi_{3120}(2287,·)$, $\chi_{3120}(1331,·)$, $\chi_{3120}(1663,·)$$\rbrace$ | ||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $\frac{1}{3} a^{2}$, $\frac{1}{3} a^{3}$, $\frac{1}{117} a^{4}$, $\frac{1}{117} a^{5}$, $\frac{1}{351} a^{6}$, $\frac{1}{351} a^{7}$, $\frac{1}{13689} a^{8}$, $\frac{1}{150579} a^{9} - \frac{1}{1287} a^{7} - \frac{4}{1287} a^{5} + \frac{1}{33} a^{3} + \frac{5}{11} a$, $\frac{1}{451737} a^{10} + \frac{5}{150579} a^{8} - \frac{4}{3861} a^{6} + \frac{2}{1287} a^{4} + \frac{5}{33} a^{2}$, $\frac{1}{451737} a^{11} - \frac{3}{11} a$, $\frac{1}{56464866315} a^{12} - \frac{74}{131619735} a^{10} - \frac{802}{43873245} a^{8} + \frac{157}{124995} a^{6} - \frac{1376}{374985} a^{4} - \frac{1889}{21153} a^{2} - \frac{241}{641}$, $\frac{1}{282324331575} a^{13} - \frac{2408}{2413028475} a^{11} - \frac{268}{268114275} a^{9} + \frac{66823}{61872525} a^{7} - \frac{5521}{20624175} a^{5} - \frac{607}{105765} a^{3} - \frac{6497}{35255} a$, $\frac{1}{889828981285090275} a^{14} - \frac{800027}{296609660428363425} a^{12} + \frac{4178621423}{7605375908419575} a^{10} - \frac{6767197546}{2535125302806525} a^{8} + \frac{87276511759}{65003212892475} a^{6} - \frac{3906114871}{1444515842055} a^{4} + \frac{1477412713}{111116603235} a^{2} - \frac{16814227}{61221269}$, $\frac{1}{889828981285090275} a^{15} + \frac{83524}{98869886809454475} a^{13} - \frac{3410905753}{7605375908419575} a^{11} + \frac{2511535547}{845041767602175} a^{9} - \frac{78221196914}{65003212892475} a^{7} + \frac{53458862681}{21667737630825} a^{5} + \frac{841373783}{22223320647} a^{3} - \frac{14772733}{3367169795} a$
Class group and class number
$C_{2}\times C_{2}\times C_{4}\times C_{8}\times C_{408}\times C_{4080}$, which has order $213073920$ (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: | \( 833353.4796729003 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| An abelian group of order 16 |
| The 16 conjugacy class representatives for $C_4^2$ |
| Character table for $C_4^2$ |
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 | ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/11.1.0.1}{1} }^{16}$ | R | ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$ |
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 | Data not computed | ||||||
| $3$ | 3.8.4.1 | $x^{8} + 36 x^{4} - 27 x^{2} + 324$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |
| 3.8.4.1 | $x^{8} + 36 x^{4} - 27 x^{2} + 324$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
| $5$ | 5.4.3.1 | $x^{4} - 5$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ |
| 5.4.3.1 | $x^{4} - 5$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ | |
| 5.4.3.1 | $x^{4} - 5$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ | |
| 5.4.3.1 | $x^{4} - 5$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ | |
| 13 | Data not computed | ||||||