Normalized defining polynomial
\( x^{16} + 680 x^{14} + 175100 x^{12} + 21732800 x^{10} + 1371644575 x^{8} + 41396938000 x^{6} + 452278497500 x^{4} + 10857730000 x^{2} + 52200625 \)
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$$\chi_{4080}(1,·)$, $\chi_{4080}(2053,·)$, $\chi_{4080}(1801,·)$, $\chi_{4080}(973,·)$, $\chi_{4080}(1871,·)$, $\chi_{4080}(2449,·)$, $\chi_{4080}(4067,·)$, $\chi_{4080}(1237,·)$, $\chi_{4080}(3671,·)$, $\chi_{4080}(1883,·)$, $\chi_{4080}(157,·)$, $\chi_{4080}(803,·)$, $\chi_{4080}(169,·)$, $\chi_{4080}(1067,·)$, $\chi_{4080}(239,·)$, $\chi_{4080}(2039,·)$$\rbrace$ | ||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{85} a^{4}$, $\frac{1}{85} a^{5}$, $\frac{1}{85} a^{6}$, $\frac{1}{85} a^{7}$, $\frac{1}{1192125} a^{8} - \frac{56}{14025} a^{6} - \frac{1}{561} a^{4} + \frac{14}{165} a^{2} + \frac{31}{165}$, $\frac{1}{1192125} a^{9} - \frac{56}{14025} a^{7} - \frac{1}{561} a^{5} + \frac{14}{165} a^{3} + \frac{31}{165} a$, $\frac{1}{1192125} a^{10} + \frac{1}{255} a^{6} + \frac{1}{15} a^{2} + \frac{10}{33}$, $\frac{1}{1192125} a^{11} + \frac{1}{255} a^{7} + \frac{1}{15} a^{3} + \frac{10}{33} a$, $\frac{1}{303991875} a^{12} + \frac{4}{3825} a^{6} - \frac{1}{425} a^{4} + \frac{47}{165} a^{2} + \frac{2}{9}$, $\frac{1}{303991875} a^{13} + \frac{4}{3825} a^{7} - \frac{1}{425} a^{5} + \frac{47}{165} a^{3} + \frac{2}{9} a$, $\frac{1}{328334936062258125} a^{14} - \frac{493942604}{328334936062258125} a^{12} + \frac{32085166}{99045229581375} a^{10} - \frac{2079127}{33589251771075} a^{8} + \frac{51633081847}{9088856361585} a^{6} + \frac{3118430437}{721337806475} a^{4} - \frac{628161580}{9720701991} a^{2} + \frac{50472483682}{106927721901}$, $\frac{1}{328334936062258125} a^{15} - \frac{493942604}{328334936062258125} a^{13} + \frac{32085166}{99045229581375} a^{11} - \frac{2079127}{33589251771075} a^{9} + \frac{51633081847}{9088856361585} a^{7} + \frac{3118430437}{721337806475} a^{5} - \frac{628161580}{9720701991} a^{3} + \frac{50472483682}{106927721901} a$
Class group and class number
$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{10}\times C_{30}\times C_{45240}$, which has order $217152000$ (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: | \( 2478557.1657538777 \) (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 | ${\href{/LocalNumberField/7.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/11.1.0.1}{1} }^{16}$ | ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ | R | ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$ |
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 | $x^{8} - 5 x^{4} + 400$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ | |
| $17$ | 17.8.6.2 | $x^{8} + 85 x^{4} + 2601$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ |
| 17.8.6.2 | $x^{8} + 85 x^{4} + 2601$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ | |