Normalized defining polynomial
\( x^{16} + 8 x^{14} + 70 x^{12} - 1034 x^{10} + 1331 x^{8} + 3298 x^{6} - 9317 x^{4} + 5213 x^{2} + 25 \)
Invariants
| Degree: | $16$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[4, 6]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(17415183620366462784744081=3^{8}\cdot 61^{12}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $37.81$ | 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: | $3, 61$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is not Galois over $\Q$. | |||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{6} a^{8} + \frac{1}{6} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a + \frac{1}{6}$, $\frac{1}{6} a^{9} + \frac{1}{6} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} + \frac{1}{6} a$, $\frac{1}{6} a^{10} + \frac{1}{6} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} + \frac{1}{6} a^{2}$, $\frac{1}{30} a^{11} + \frac{1}{30} a^{9} + \frac{7}{30} a^{7} - \frac{7}{15} a^{5} - \frac{1}{2} a^{4} - \frac{4}{15} a^{3} - \frac{1}{2} a^{2} + \frac{13}{30} a$, $\frac{1}{90} a^{12} + \frac{1}{90} a^{10} + \frac{1}{45} a^{8} + \frac{8}{45} a^{6} - \frac{1}{2} a^{5} + \frac{17}{90} a^{4} - \frac{1}{2} a^{3} - \frac{1}{45} a^{2} - \frac{1}{2} a - \frac{7}{18}$, $\frac{1}{90} a^{13} + \frac{1}{90} a^{11} + \frac{1}{45} a^{9} + \frac{8}{45} a^{7} - \frac{1}{2} a^{6} + \frac{17}{90} a^{5} - \frac{1}{2} a^{4} - \frac{1}{45} a^{3} - \frac{1}{2} a^{2} - \frac{7}{18} a$, $\frac{1}{3321812925450} a^{14} - \frac{72784982}{61515054175} a^{12} - \frac{1331659657}{22752143325} a^{10} - \frac{8956213421}{1660906462725} a^{8} - \frac{1}{2} a^{7} + \frac{597310438634}{1660906462725} a^{6} + \frac{1391954368}{66436258509} a^{4} - \frac{1}{2} a^{3} + \frac{15805049356}{184545162525} a^{2} - \frac{1}{2} a - \frac{27460654633}{132872517018}$, $\frac{1}{3321812925450} a^{15} - \frac{72784982}{61515054175} a^{13} + \frac{185149898}{22752143325} a^{11} + \frac{101770884094}{1660906462725} a^{9} - \frac{288506341486}{1660906462725} a^{7} + \frac{29105191343}{332181292545} a^{5} - \frac{27539679108}{61515054175} a^{3} + \frac{53148170684}{332181292545} a - \frac{1}{2}$
Class group and class number
$C_{3}$, which has order $3$ (assuming GRH)
Unit group
| Rank: | $9$ | 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: | \( 3899874.07927 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A solvable group of order 32 |
| The 14 conjugacy class representatives for $D_4:C_4$ |
| Character table for $D_4:C_4$ |
Intermediate fields
| \(\Q(\sqrt{61}) \), 4.4.2042829.1, 4.2.11163.1, 4.2.680943.1, 8.2.373837707.1, 8.2.1391050107747.2, 8.4.4173150323241.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Galois closure: | data not computed |
| Degree 16 sibling: | data not computed |
Frobenius cycle types
| $p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 | 53 | 59 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ | R | ${\href{/LocalNumberField/5.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{4}$ | ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/41.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ | ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
| $3$ | 3.2.1.1 | $x^{2} - 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 3.2.1.1 | $x^{2} - 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 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 | $x^{4} + 9 x^{2} + 36$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| $61$ | 61.8.6.1 | $x^{8} - 61 x^{4} + 59536$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ |
| 61.8.6.1 | $x^{8} - 61 x^{4} + 59536$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ |