Normalized defining polynomial
\( x^{16} + 816 x^{14} + 272952 x^{12} + 48191328 x^{10} + 4808091555 x^{8} + 267911510472 x^{6} + 7592642387532 x^{4} + 82436842630224 x^{2} + 31301238752001 \)
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: | \(262686006513622818702917369856000000000000=2^{48}\cdot 3^{8}\cdot 5^{12}\cdot 17^{12}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $387.90$ | 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: | $2, 3, 5, 17$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(4080=2^{4}\cdot 3\cdot 5\cdot 17\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{4080}(1,·)$, $\chi_{4080}(1801,·)$, $\chi_{4080}(2189,·)$, $\chi_{4080}(2449,·)$, $\chi_{4080}(3923,·)$, $\chi_{4080}(1109,·)$, $\chi_{4080}(2843,·)$, $\chi_{4080}(2143,·)$, $\chi_{4080}(3107,·)$, $\chi_{4080}(3943,·)$, $\chi_{4080}(169,·)$, $\chi_{4080}(2027,·)$, $\chi_{4080}(3821,·)$, $\chi_{4080}(1327,·)$, $\chi_{4080}(2741,·)$, $\chi_{4080}(3127,·)$$\rbrace$ | ||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $\frac{1}{3} a^{2}$, $\frac{1}{3} a^{3}$, $\frac{1}{153} a^{4}$, $\frac{1}{153} a^{5}$, $\frac{1}{459} a^{6}$, $\frac{1}{459} a^{7}$, $\frac{1}{351135} a^{8} - \frac{7}{6885} a^{6} + \frac{1}{765} a^{4} + \frac{4}{45} a^{2} - \frac{1}{15}$, $\frac{1}{351135} a^{9} - \frac{7}{6885} a^{7} + \frac{1}{765} a^{5} + \frac{4}{45} a^{3} - \frac{1}{15} a$, $\frac{1}{321288525} a^{10} - \frac{1}{793305} a^{8} - \frac{64}{419985} a^{6} - \frac{38}{139995} a^{4} - \frac{19}{2745} a^{2} - \frac{2069}{4575}$, $\frac{1}{76787957475} a^{11} + \frac{949}{5119197165} a^{9} - \frac{64541}{100376415} a^{7} - \frac{2899}{6691761} a^{5} + \frac{6908}{43737} a^{3} - \frac{482749}{1093425} a$, $\frac{1}{3916185831225} a^{12} + \frac{4}{25595985825} a^{10} - \frac{728}{5119197165} a^{8} - \frac{256}{3717645} a^{6} - \frac{2459}{6691761} a^{4} + \frac{25392}{364475} a^{2} - \frac{1832}{4575}$, $\frac{1}{3916185831225} a^{13} + \frac{821}{1706399055} a^{9} + \frac{9472}{100376415} a^{7} - \frac{13303}{33458805} a^{5} - \frac{229967}{3280275} a^{3} - \frac{36976}{218685} a$, $\frac{1}{11748557493675} a^{14} - \frac{113}{76787957475} a^{10} + \frac{298}{1706399055} a^{8} + \frac{1196}{1239215} a^{6} - \frac{3869}{2742525} a^{4} - \frac{34108}{656055} a^{2} - \frac{992}{4575}$, $\frac{1}{11748557493675} a^{15} + \frac{2026}{1706399055} a^{9} + \frac{78664}{100376415} a^{7} - \frac{115654}{167294025} a^{5} - \frac{6992}{43737} a^{3} + \frac{78758}{218685} a$
Class group and class number
$C_{2}\times C_{4}\times C_{4}\times C_{4}\times C_{24}\times C_{45240}$, which has order $138977280$ (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: | \( 663170.015206084 \) (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 | ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ | R | ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$ |
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 | $x^{8} + 36 x^{4} - 27 x^{2} + 324$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
| $5$ | 5.8.6.1 | $x^{8} - 5 x^{4} + 400$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ |
| 5.8.6.1 | $x^{8} - 5 x^{4} + 400$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ | |
| 17 | Data not computed | ||||||