Normalized defining polynomial
\( x^{16} + 472 x^{14} + 88780 x^{12} + 8569912 x^{10} + 456984038 x^{8} + 13450643240 x^{6} + 205334750052 x^{4} + 1420420813840 x^{2} + 3471343735202 \)
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: | \(21393065423814767306817757436903948288=2^{67}\cdot 7^{2}\cdot 17^{6}\cdot 11071^{2}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $215.35$ | 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: | $2, 7, 17, 11071$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is not Galois over $\Q$. | |||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{7} a^{12} + \frac{1}{7} a^{10} - \frac{3}{7} a^{8} - \frac{1}{7} a^{2}$, $\frac{1}{7} a^{13} + \frac{1}{7} a^{11} - \frac{3}{7} a^{9} - \frac{1}{7} a^{3}$, $\frac{1}{2343197010653074124796798686633678585321} a^{14} - \frac{86593489284293478604323882852871023377}{2343197010653074124796798686633678585321} a^{12} - \frac{421771715170546232913281876566061387985}{2343197010653074124796798686633678585321} a^{10} - \frac{97593609474573200164395733641460905953}{334742430093296303542399812376239797903} a^{8} + \frac{7638593258721495096037089626954570172}{19690731181958606090729400728014105759} a^{6} - \frac{757555186043215340668823879098006684533}{2343197010653074124796798686633678585321} a^{4} - \frac{44529765688488733543776058614613209742}{334742430093296303542399812376239797903} a^{2} + \frac{636499753763125510789526043462421}{1778586503654467174666191015085729}$, $\frac{1}{2343197010653074124796798686633678585321} a^{15} - \frac{86593489284293478604323882852871023377}{2343197010653074124796798686633678585321} a^{13} - \frac{421771715170546232913281876566061387985}{2343197010653074124796798686633678585321} a^{11} - \frac{97593609474573200164395733641460905953}{334742430093296303542399812376239797903} a^{9} + \frac{7638593258721495096037089626954570172}{19690731181958606090729400728014105759} a^{7} - \frac{757555186043215340668823879098006684533}{2343197010653074124796798686633678585321} a^{5} - \frac{44529765688488733543776058614613209742}{334742430093296303542399812376239797903} a^{3} + \frac{636499753763125510789526043462421}{1778586503654467174666191015085729} a$
Class group and class number
$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{1501776}$, which has order $24028416$ (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: | \( 42439.7060581 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A solvable group of order 1024 |
| The 49 conjugacy class representatives for t16n1113 |
| Character table for t16n1113 is not computed |
Intermediate fields
| \(\Q(\sqrt{2}) \), 4.4.4352.1, 8.8.9697230848.1 |
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 | R | $16$ | $16$ | R | $16$ | ${\href{/LocalNumberField/13.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ | R | ${\href{/LocalNumberField/19.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ | $16$ | ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ | $16$ | ${\href{/LocalNumberField/41.8.0.1}{8} }{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/59.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
| 2 | Data not computed | ||||||
| $7$ | 7.2.0.1 | $x^{2} - x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 7.2.0.1 | $x^{2} - x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 7.4.2.2 | $x^{4} - 7 x^{2} + 147$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ | |
| 7.8.0.1 | $x^{8} - x + 3$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | |
| $17$ | 17.2.0.1 | $x^{2} - x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 17.2.0.1 | $x^{2} - x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 17.4.2.1 | $x^{4} + 85 x^{2} + 2601$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 17.4.2.1 | $x^{4} + 85 x^{2} + 2601$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 17.4.2.1 | $x^{4} + 85 x^{2} + 2601$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 11071 | Data not computed | ||||||