Normalized defining polynomial
\( x^{16} - 30 x^{14} + 125 x^{12} - 60 x^{10} + 8510 x^{8} - 41150 x^{6} + 77600 x^{4} - 66400 x^{2} + 21025 \)
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: | \(1739116855398400000000000000=2^{24}\cdot 5^{14}\cdot 19^{8}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $50.41$ | 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: | $2, 5, 19$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{10} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2}$, $\frac{1}{10} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{610} a^{10} + \frac{57}{122} a^{6} - \frac{3}{61} a^{4} - \frac{33}{122} a^{2} + \frac{49}{122}$, $\frac{1}{610} a^{11} + \frac{57}{122} a^{7} - \frac{3}{61} a^{5} - \frac{33}{122} a^{3} + \frac{49}{122} a$, $\frac{1}{1220} a^{12} - \frac{1}{1220} a^{11} - \frac{1}{1220} a^{10} - \frac{1}{20} a^{9} + \frac{41}{1220} a^{8} - \frac{59}{122} a^{7} + \frac{59}{244} a^{6} + \frac{3}{122} a^{5} + \frac{95}{244} a^{4} - \frac{89}{244} a^{3} - \frac{10}{61} a^{2} + \frac{3}{61} a + \frac{73}{244}$, $\frac{1}{1220} a^{13} - \frac{1}{61} a^{9} - \frac{1}{20} a^{8} + \frac{55}{244} a^{7} - \frac{1}{4} a^{6} + \frac{89}{244} a^{5} - \frac{1}{2} a^{4} + \frac{49}{244} a^{3} - \frac{1}{2} a^{2} - \frac{1}{4} a - \frac{1}{4}$, $\frac{1}{21797958380} a^{14} + \frac{699781}{1981632580} a^{12} - \frac{1}{1220} a^{11} + \frac{15874467}{21797958380} a^{10} - \frac{91898823}{2179795838} a^{8} - \frac{57}{244} a^{7} - \frac{120947988}{1089897919} a^{6} - \frac{29}{61} a^{5} + \frac{691231483}{2179795838} a^{4} + \frac{33}{244} a^{3} - \frac{1827845079}{4359591676} a^{2} + \frac{73}{244} a + \frac{1533386713}{4359591676}$, $\frac{1}{3160703965100} a^{15} - \frac{6264749}{28733672410} a^{13} - \frac{110975781}{316070396510} a^{11} - \frac{1}{1220} a^{10} - \frac{18104578183}{632140793020} a^{9} - \frac{1}{20} a^{8} + \frac{259966076331}{632140793020} a^{7} - \frac{59}{122} a^{6} - \frac{31702184381}{126428158604} a^{5} + \frac{3}{122} a^{4} - \frac{13279965997}{126428158604} a^{3} - \frac{89}{244} a^{2} + \frac{28424576003}{63214079302} a + \frac{3}{61}$
Class group and class number
$C_{2}\times C_{2}$, which has order $4$ (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: | \( 17951375.8602 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
$C_2^3.C_4$ (as 16T36):
| A solvable group of order 32 |
| The 11 conjugacy class representatives for $C_2^3.C_4$ |
| Character table for $C_2^3.C_4$ |
Intermediate fields
| \(\Q(\sqrt{5}) \), \(\Q(\sqrt{95}) \), \(\Q(\sqrt{19}) \), 4.4.45125.1, \(\Q(\zeta_{20})^+\), \(\Q(\sqrt{5}, \sqrt{19})\), 8.4.2606420000000.2 x2, 8.4.115520000000.2 x2, 8.8.521284000000.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Galois closure: | data not computed |
| Degree 8 siblings: | data not computed |
| Degree 16 siblings: | data not computed |
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 | ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ | R | ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$ |
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 | ||||||
| $5$ | 5.8.7.2 | $x^{8} - 20$ | $8$ | $1$ | $7$ | $C_8:C_2$ | $[\ ]_{8}^{2}$ |
| 5.8.7.2 | $x^{8} - 20$ | $8$ | $1$ | $7$ | $C_8:C_2$ | $[\ ]_{8}^{2}$ | |
| $19$ | 19.2.1.1 | $x^{2} - 19$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 19.2.1.1 | $x^{2} - 19$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 19.2.1.1 | $x^{2} - 19$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 19.2.1.1 | $x^{2} - 19$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 19.4.2.1 | $x^{4} + 57 x^{2} + 1444$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 19.4.2.1 | $x^{4} + 57 x^{2} + 1444$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |