Normalized defining polynomial
\( x^{16} - 4 x^{15} + 8 x^{14} - 13 x^{13} + 64 x^{12} - 283 x^{11} + 576 x^{10} - 589 x^{9} - 105 x^{8} + 766 x^{7} - 531 x^{6} - 275 x^{5} + 547 x^{4} + 139 x^{3} - 79 x^{2} + 34 x - 11 \)
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: | \(5226532666677246533203125=3^{6}\cdot 5^{11}\cdot 59^{8}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $35.07$ | 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: | $3, 5, 59$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{31} a^{14} + \frac{15}{31} a^{13} - \frac{8}{31} a^{12} + \frac{1}{31} a^{11} + \frac{11}{31} a^{10} - \frac{3}{31} a^{9} - \frac{2}{31} a^{8} - \frac{3}{31} a^{7} + \frac{6}{31} a^{6} - \frac{15}{31} a^{5} + \frac{13}{31} a^{4} - \frac{8}{31} a^{3} - \frac{15}{31} a^{2} - \frac{1}{31} a + \frac{15}{31}$, $\frac{1}{8580459435425562559} a^{15} + \frac{39334718698344122}{8580459435425562559} a^{14} + \frac{400067724032050541}{8580459435425562559} a^{13} - \frac{59050107463934280}{8580459435425562559} a^{12} + \frac{550190139126447386}{1225779919346508937} a^{11} + \frac{286168079700670381}{8580459435425562559} a^{10} + \frac{4158431564553515122}{8580459435425562559} a^{9} - \frac{2214293919050058736}{8580459435425562559} a^{8} - \frac{3101069498723771304}{8580459435425562559} a^{7} + \frac{1847704314174368651}{8580459435425562559} a^{6} - \frac{445715077740607736}{8580459435425562559} a^{5} + \frac{385248939518666123}{1225779919346508937} a^{4} - \frac{272384356641917309}{8580459435425562559} a^{3} - \frac{16209592273129294}{1225779919346508937} a^{2} + \frac{2928674400225862076}{8580459435425562559} a + \frac{2085730637181548452}{8580459435425562559}$
Class group and class number
Trivial group, which has order $1$ (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: | \( 3361271.16649 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A solvable group of order 512 |
| The 32 conjugacy class representatives for t16n937 |
| Character table for t16n937 is not computed |
Intermediate fields
| \(\Q(\sqrt{5}) \), 4.2.1475.1, 8.4.340800778125.1 |
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 | $16$ | R | R | ${\href{/LocalNumberField/7.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{4}$ | $16$ | ${\href{/LocalNumberField/17.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ | $16$ | ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ | $16$ | ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ | $16$ | $16$ | ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ | 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 |
|---|---|---|---|---|---|---|---|
| $3$ | 3.4.2.2 | $x^{4} - 3 x^{2} + 18$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ |
| 3.4.2.2 | $x^{4} - 3 x^{2} + 18$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ | |
| 3.4.0.1 | $x^{4} - x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 3.4.2.1 | $x^{4} + 9 x^{2} + 36$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| $5$ | 5.4.3.1 | $x^{4} - 5$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ |
| 5.4.3.1 | $x^{4} - 5$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ | |
| 5.4.2.1 | $x^{4} + 15 x^{2} + 100$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 5.4.3.2 | $x^{4} - 20$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ | |
| $59$ | 59.2.1.2 | $x^{2} + 177$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 59.2.1.2 | $x^{2} + 177$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 59.2.1.2 | $x^{2} + 177$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 59.2.1.2 | $x^{2} + 177$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 59.8.4.1 | $x^{8} + 97468 x^{4} - 205379 x^{2} + 2375002756$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |