Normalized defining polynomial
\( x^{16} - 40 x^{14} + 588 x^{12} - 3926 x^{10} + 12244 x^{8} - 11204 x^{6} + 21321 x^{4} - 5764 x^{2} + 1936 \)
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: | \(31634849063620633600000000=2^{16}\cdot 5^{8}\cdot 7^{8}\cdot 11^{8}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $39.24$ | 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: | $2, 5, 7, 11$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(1540=2^{2}\cdot 5\cdot 7\cdot 11\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{1540}(1,·)$, $\chi_{1540}(1539,·)$, $\chi_{1540}(769,·)$, $\chi_{1540}(461,·)$, $\chi_{1540}(1231,·)$, $\chi_{1540}(659,·)$, $\chi_{1540}(1429,·)$, $\chi_{1540}(351,·)$, $\chi_{1540}(1121,·)$, $\chi_{1540}(419,·)$, $\chi_{1540}(1189,·)$, $\chi_{1540}(111,·)$, $\chi_{1540}(881,·)$, $\chi_{1540}(309,·)$, $\chi_{1540}(1079,·)$, $\chi_{1540}(771,·)$$\rbrace$ | ||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{12} a^{8} - \frac{1}{4} a^{2} - \frac{1}{3}$, $\frac{1}{12} a^{9} - \frac{1}{4} a^{3} - \frac{1}{3} a$, $\frac{1}{12} a^{10} - \frac{1}{4} a^{4} - \frac{1}{3} a^{2}$, $\frac{1}{12} a^{11} - \frac{1}{4} a^{5} - \frac{1}{3} a^{3}$, $\frac{1}{3780} a^{12} - \frac{37}{1260} a^{10} - \frac{23}{756} a^{8} + \frac{31}{1260} a^{6} - \frac{79}{3780} a^{4} + \frac{137}{420} a^{2} + \frac{361}{945}$, $\frac{1}{7560} a^{13} - \frac{37}{2520} a^{11} - \frac{23}{1512} a^{9} + \frac{31}{2520} a^{7} - \frac{79}{7560} a^{5} - \frac{283}{840} a^{3} + \frac{361}{1890} a$, $\frac{1}{184460771880} a^{14} - \frac{69835}{1190069496} a^{12} - \frac{896716363}{26351538840} a^{10} + \frac{850381199}{26351538840} a^{8} + \frac{63145457519}{184460771880} a^{6} - \frac{1999711621}{184460771880} a^{4} - \frac{5039789701}{23057596485} a^{2} - \frac{795250997}{2096145135}$, $\frac{1}{2029068490680} a^{15} - \frac{69835}{13090764456} a^{13} + \frac{3495206777}{289866927240} a^{11} + \frac{9634227479}{289866927240} a^{9} + \frac{432067001279}{2029068490680} a^{7} + \frac{1012534533719}{2029068490680} a^{5} - \frac{89584310146}{253633561335} a^{3} - \frac{1493966042}{23057596485} a$
Class group and class number
$C_{4}\times C_{4}$, which has order $16$ (assuming GRH)
Unit group
| Rank: | $7$ | magma: UnitRank(K);
sage: UK.rank()
gp: K.fu
| |
| Torsion generator: | \( -\frac{98161}{279524520} a^{15} + \frac{31409}{2254230} a^{13} - \frac{1410674}{6988113} a^{11} + \frac{182928071}{139762260} a^{9} - \frac{266402369}{69881130} a^{7} + \frac{80537474}{34940565} a^{5} - \frac{1576600489}{279524520} a^{3} - \frac{85532}{635283} a \) (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: | \( 696645.837098 \) (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 $C_2^4$ |
| Character table for $C_2^4$ |
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 | ${\href{/LocalNumberField/3.2.0.1}{2} }^{8}$ | R | R | R | ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$ |
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 |
|---|---|---|---|---|---|---|---|
| $2$ | 2.4.4.1 | $x^{4} + 8 x^{2} + 4$ | $2$ | $2$ | $4$ | $C_2^2$ | $[2]^{2}$ |
| 2.4.4.1 | $x^{4} + 8 x^{2} + 4$ | $2$ | $2$ | $4$ | $C_2^2$ | $[2]^{2}$ | |
| 2.4.4.1 | $x^{4} + 8 x^{2} + 4$ | $2$ | $2$ | $4$ | $C_2^2$ | $[2]^{2}$ | |
| 2.4.4.1 | $x^{4} + 8 x^{2} + 4$ | $2$ | $2$ | $4$ | $C_2^2$ | $[2]^{2}$ | |
| $5$ | 5.4.2.1 | $x^{4} + 15 x^{2} + 100$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
| 5.4.2.1 | $x^{4} + 15 x^{2} + 100$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 5.4.2.1 | $x^{4} + 15 x^{2} + 100$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 5.4.2.1 | $x^{4} + 15 x^{2} + 100$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| $7$ | 7.4.2.1 | $x^{4} + 35 x^{2} + 441$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
| 7.4.2.1 | $x^{4} + 35 x^{2} + 441$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 7.4.2.1 | $x^{4} + 35 x^{2} + 441$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 7.4.2.1 | $x^{4} + 35 x^{2} + 441$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| $11$ | 11.4.2.1 | $x^{4} + 143 x^{2} + 5929$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
| 11.4.2.1 | $x^{4} + 143 x^{2} + 5929$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 11.4.2.1 | $x^{4} + 143 x^{2} + 5929$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 11.4.2.1 | $x^{4} + 143 x^{2} + 5929$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |