Normalized defining polynomial
\( x^{16} - 6 x^{15} - 23 x^{14} + 76 x^{13} - 214 x^{12} + 1002 x^{11} + 7538 x^{10} - 10546 x^{9} - 15990 x^{8} + 3618 x^{7} - 308637 x^{6} - 190596 x^{5} + 65402 x^{4} - 242146 x^{3} + 1530257 x^{2} - 1648578 x - 238999 \)
Invariants
| Degree: | $16$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[8, 4]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(2800444573576914534400000000=2^{24}\cdot 5^{8}\cdot 13^{6}\cdot 97^{4}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $51.93$ | 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: | $2, 5, 13, 97$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{2861} a^{14} - \frac{914}{2861} a^{13} + \frac{816}{2861} a^{12} - \frac{174}{2861} a^{11} + \frac{359}{2861} a^{10} - \frac{317}{2861} a^{9} - \frac{964}{2861} a^{8} - \frac{297}{2861} a^{7} - \frac{643}{2861} a^{6} + \frac{812}{2861} a^{5} - \frac{720}{2861} a^{4} + \frac{11}{2861} a^{3} + \frac{270}{2861} a^{2} + \frac{129}{2861} a + \frac{443}{2861}$, $\frac{1}{309375804049599998294179518069220359472390570369} a^{15} + \frac{6593238996675735926481164514092844934123740}{309375804049599998294179518069220359472390570369} a^{14} + \frac{113950543060811318752747846524844202602845292112}{309375804049599998294179518069220359472390570369} a^{13} - \frac{63391518590249693200838384952430978247616307451}{309375804049599998294179518069220359472390570369} a^{12} - \frac{47897176668630313930561901096640728279326453865}{309375804049599998294179518069220359472390570369} a^{11} - \frac{52712499384280958056889968709199561436956531885}{309375804049599998294179518069220359472390570369} a^{10} - \frac{71517231978291369609416219721393777720697384971}{309375804049599998294179518069220359472390570369} a^{9} - \frac{55330547759488934806907046079869574107764466994}{309375804049599998294179518069220359472390570369} a^{8} + \frac{139189160548586979454568097327561530717552757392}{309375804049599998294179518069220359472390570369} a^{7} + \frac{4307185787691786418104322511898483741929460861}{9979864646761290267554178002232914821690018399} a^{6} - \frac{4092239389811385661053241103621264259520881416}{309375804049599998294179518069220359472390570369} a^{5} + \frac{143677768352877476753090882738522144063634362140}{309375804049599998294179518069220359472390570369} a^{4} + \frac{138141118429865023784246990521716665346651257439}{309375804049599998294179518069220359472390570369} a^{3} - \frac{122132474930742380727731152927808516242166683814}{309375804049599998294179518069220359472390570369} a^{2} - \frac{51869341111645250160125825952111785046478348315}{309375804049599998294179518069220359472390570369} a - \frac{56016113223792998066628932239081836331807304464}{309375804049599998294179518069220359472390570369}$
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
| Rank: | $11$ | 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: | \( 49460085.7949 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
$(C_2\times C_4).C_2^4$ (as 16T205):
| A solvable group of order 128 |
| The 44 conjugacy class representatives for $(C_2\times C_4).C_2^4$ |
| Character table for $(C_2\times C_4).C_2^4$ is not computed |
Intermediate fields
| \(\Q(\sqrt{5}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{10}) \), \(\Q(\sqrt{2}, \sqrt{5})\), 8.8.432640000.1 |
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 | ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ | R | ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ | R | ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{8}$ | ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ | ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
| 2 | Data not computed | ||||||
| 5 | Data not computed | ||||||
| $13$ | 13.8.0.1 | $x^{8} + 4 x^{2} - x + 6$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ |
| 13.8.6.3 | $x^{8} + 65 x^{4} + 1352$ | $4$ | $2$ | $6$ | $C_8$ | $[\ ]_{4}^{2}$ | |
| $97$ | 97.8.0.1 | $x^{8} - x + 84$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ |
| 97.8.4.2 | $x^{8} - 912673 x^{2} + 2036173463$ | $2$ | $4$ | $4$ | $C_8$ | $[\ ]_{2}^{4}$ | |