Normalized defining polynomial
\( x^{16} - 8 x^{15} + 22 x^{14} - 14 x^{13} - 15 x^{12} - 92 x^{11} + 164 x^{10} + 643 x^{9} - 1478 x^{8} - 102 x^{7} - 391 x^{6} + 6462 x^{5} - 7540 x^{4} + 1781 x^{3} + 761 x^{2} - 194 x + 113 \)
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: | \(10510588883992212332738001=3^{6}\cdot 7^{2}\cdot 17^{8}\cdot 59^{6}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $36.63$ | 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: | $3, 7, 17, 59$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{3} a^{12} + \frac{1}{3} a^{10} - \frac{1}{3} a^{9} - \frac{1}{3} a^{6} + \frac{1}{3} a^{4} - \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{3} a^{13} + \frac{1}{3} a^{11} - \frac{1}{3} a^{10} - \frac{1}{3} a^{7} + \frac{1}{3} a^{5} - \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{115832895114} a^{14} - \frac{7}{115832895114} a^{13} + \frac{1439668852}{19305482519} a^{12} - \frac{51828078581}{115832895114} a^{11} - \frac{1482047131}{19305482519} a^{10} + \frac{56220553633}{115832895114} a^{9} - \frac{53321497429}{115832895114} a^{8} + \frac{851174497}{115832895114} a^{7} - \frac{23165719307}{57916447557} a^{6} - \frac{7579345483}{115832895114} a^{5} - \frac{15132351842}{57916447557} a^{4} - \frac{53064097291}{115832895114} a^{3} + \frac{16033996349}{38610965038} a^{2} + \frac{21636818471}{115832895114} a + \frac{21699827155}{115832895114}$, $\frac{1}{361051134070338} a^{15} + \frac{517}{120350378023446} a^{14} - \frac{29146959602587}{180525567035169} a^{13} + \frac{3480187125283}{40116792674482} a^{12} + \frac{1051781241451}{60175189011723} a^{11} - \frac{147700782778739}{361051134070338} a^{10} - \frac{12256376459}{347498685342} a^{9} + \frac{14085716897821}{120350378023446} a^{8} + \frac{58170237120476}{180525567035169} a^{7} - \frac{42731794382101}{361051134070338} a^{6} + \frac{7377655003675}{20058396337241} a^{5} + \frac{5767652757287}{40116792674482} a^{4} + \frac{94714400829203}{361051134070338} a^{3} - \frac{11949943997881}{40116792674482} a^{2} + \frac{68790619259477}{361051134070338} a + \frac{63314545329421}{180525567035169}$
Class group and class number
$C_{2}$, which has order $2$
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 | magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
| |
| Regulator: | \( 2276889.47545 \) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A solvable group of order 3072 |
| The 60 conjugacy class representatives for t16n1537 are not computed |
| Character table for t16n1537 is not computed |
Intermediate fields
| 4.4.51153.1, 8.2.3242003837751.1, 8.8.463143405393.1, 8.2.18316405863.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Frobenius cycle types
| $p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 | 53 | 59 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/LocalNumberField/2.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/2.2.0.1}{2} }^{2}$ | R | ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ | R | ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ | R | ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ | ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/47.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ | R |
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 |
|---|---|---|---|---|---|---|---|
| $3$ | 3.4.0.1 | $x^{4} - x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |
| 3.4.3.2 | $x^{4} - 3$ | $4$ | $1$ | $3$ | $D_{4}$ | $[\ ]_{4}^{2}$ | |
| 3.4.3.2 | $x^{4} - 3$ | $4$ | $1$ | $3$ | $D_{4}$ | $[\ ]_{4}^{2}$ | |
| 3.4.0.1 | $x^{4} - x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| $7$ | 7.4.2.1 | $x^{4} + 35 x^{2} + 441$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
| 7.6.0.1 | $x^{6} + 3 x^{2} - x + 5$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
| 7.6.0.1 | $x^{6} + 3 x^{2} - x + 5$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
| $17$ | 17.2.0.1 | $x^{2} - x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 17.2.0.1 | $x^{2} - x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 17.6.4.1 | $x^{6} + 136 x^{3} + 7803$ | $3$ | $2$ | $4$ | $S_3$ | $[\ ]_{3}^{2}$ | |
| 17.6.4.1 | $x^{6} + 136 x^{3} + 7803$ | $3$ | $2$ | $4$ | $S_3$ | $[\ ]_{3}^{2}$ | |
| $59$ | 59.4.0.1 | $x^{4} - x + 14$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |
| 59.4.0.1 | $x^{4} - x + 14$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 59.8.6.1 | $x^{8} - 59 x^{4} + 55696$ | $4$ | $2$ | $6$ | $D_4$ | $[\ ]_{4}^{2}$ |