Normalized defining polynomial
\( x^{16} - 336 x^{14} + 29588 x^{12} + 540680 x^{10} - 144580170 x^{8} + 2481964568 x^{6} + 85749047868 x^{4} - 2645562716896 x^{2} + 18683736527618 \)
Invariants
| Degree: | $16$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[12, 2]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(115143422370815907323395348998014369792=2^{67}\cdot 17^{6}\cdot 23^{2}\cdot 7817^{2}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $239.24$ | 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: | $2, 17, 23, 7817$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{555714395635577798742691633473762038049986713} a^{14} + \frac{117173252713999547275056329370516337696418138}{555714395635577798742691633473762038049986713} a^{12} + \frac{2624043363146941208079205918664039766419565}{79387770805082542677527376210537434007140959} a^{10} + \frac{10864518931210184714348052604308750559372607}{79387770805082542677527376210537434007140959} a^{8} + \frac{23369460969358496430543779445670460115000436}{79387770805082542677527376210537434007140959} a^{6} - \frac{267198400216894354171182660891686981656435658}{555714395635577798742691633473762038049986713} a^{4} - \frac{18839131754568649374212315555085149675508}{6106751600390964821348259708502879539010843} a^{2} - \frac{78598518354091294021138781037671319648}{181817121525607281507806820623345354279}$, $\frac{1}{555714395635577798742691633473762038049986713} a^{15} + \frac{117173252713999547275056329370516337696418138}{555714395635577798742691633473762038049986713} a^{13} + \frac{2624043363146941208079205918664039766419565}{79387770805082542677527376210537434007140959} a^{11} + \frac{10864518931210184714348052604308750559372607}{79387770805082542677527376210537434007140959} a^{9} + \frac{23369460969358496430543779445670460115000436}{79387770805082542677527376210537434007140959} a^{7} - \frac{267198400216894354171182660891686981656435658}{555714395635577798742691633473762038049986713} a^{5} - \frac{18839131754568649374212315555085149675508}{6106751600390964821348259708502879539010843} a^{3} - \frac{78598518354091294021138781037671319648}{181817121525607281507806820623345354279} a$
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
| Rank: | $13$ | 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: | \( 81124052995300 \) (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 | $16$ | $16$ | ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{4}$ | $16$ | ${\href{/LocalNumberField/13.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ | R | ${\href{/LocalNumberField/19.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ | R | $16$ | ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ | $16$ | ${\href{/LocalNumberField/41.8.0.1}{8} }{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ | ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/59.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$ |
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 | ||||||
| $17$ | 17.4.2.2 | $x^{4} - 17 x^{2} + 867$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ |
| 17.4.0.1 | $x^{4} - x + 11$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 17.4.2.1 | $x^{4} + 85 x^{2} + 2601$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 17.4.2.1 | $x^{4} + 85 x^{2} + 2601$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| $23$ | 23.2.1.2 | $x^{2} + 46$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 23.2.1.2 | $x^{2} + 46$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 23.4.0.1 | $x^{4} - x + 11$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 23.8.0.1 | $x^{8} + x^{2} - 2 x + 5$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | |
| 7817 | Data not computed | ||||||