Normalized defining polynomial
\( x^{8} - 4 x^{7} + 24 x^{6} - 58 x^{5} + 811 x^{4} - 1530 x^{3} + 12626 x^{2} - 11870 x + 115355 \)
Invariants
| Degree: | $8$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[0, 4]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(1127752403100625=5^{4}\cdot 19^{4}\cdot 61^{4}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $76.12$ | 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: | $5, 19, 61$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois over $\Q$. | |||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $\frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{32} a^{4} - \frac{1}{16} a^{3} + \frac{5}{32} a^{2} - \frac{1}{8} a + \frac{11}{32}$, $\frac{1}{32} a^{5} + \frac{1}{32} a^{3} + \frac{3}{16} a^{2} + \frac{3}{32} a - \frac{5}{16}$, $\frac{1}{2560} a^{6} - \frac{3}{2560} a^{5} - \frac{9}{1280} a^{4} + \frac{41}{2560} a^{3} + \frac{137}{1280} a^{2} - \frac{59}{512} a + \frac{73}{512}$, $\frac{1}{2746880} a^{7} + \frac{533}{2746880} a^{6} - \frac{10853}{1373440} a^{5} - \frac{2327}{2746880} a^{4} - \frac{68439}{274688} a^{3} + \frac{383769}{2746880} a^{2} + \frac{102385}{549376} a - \frac{26423}{68672}$
Class group and class number
$C_{2}\times C_{6}\times C_{96}$, which has order $1152$
Unit group
| Rank: | $3$ | 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 | magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
| |
| Regulator: | \( 71.6628735611 \) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A solvable group of order 8 |
| The 5 conjugacy class representatives for $D_4$ |
| Character table for $D_4$ |
Intermediate fields
| \(\Q(\sqrt{-5795}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-1159}) \), \(\Q(\sqrt{5}, \sqrt{-1159})\), 4.2.28975.2 x2, 4.0.6716405.1 x2 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Degree 4 siblings: | 4.2.28975.2, 4.0.6716405.1 |
Frobenius cycle types
| $p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 | 53 | 59 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/LocalNumberField/2.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/3.4.0.1}{4} }^{2}$ | R | ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}$ | ${\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}$ | ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}$ | R | ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}$ | ${\href{/LocalNumberField/29.1.0.1}{1} }^{8}$ | ${\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}$ | ${\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/59.2.0.1}{2} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
| $5$ | 5.2.1.1 | $x^{2} - 5$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 5.2.1.1 | $x^{2} - 5$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 5.2.1.1 | $x^{2} - 5$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 5.2.1.1 | $x^{2} - 5$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| $19$ | 19.2.1.2 | $x^{2} + 76$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 19.2.1.2 | $x^{2} + 76$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 19.2.1.2 | $x^{2} + 76$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 19.2.1.2 | $x^{2} + 76$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| $61$ | 61.2.1.1 | $x^{2} - 61$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 61.2.1.1 | $x^{2} - 61$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 61.2.1.1 | $x^{2} - 61$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 61.2.1.1 | $x^{2} - 61$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
Artin representations
| Label | Dimension | Conductor | Defining polynomial of Artin field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| * | 1.1.1t1.1c1 | $1$ | $1$ | $x$ | $C_1$ | $1$ | $1$ |
| * | 1.19_61.2t1.1c1 | $1$ | $ 19 \cdot 61 $ | $x^{2} - x + 290$ | $C_2$ (as 2T1) | $1$ | $-1$ |
| * | 1.5.2t1.1c1 | $1$ | $ 5 $ | $x^{2} - x - 1$ | $C_2$ (as 2T1) | $1$ | $1$ |
| * | 1.5_19_61.2t1.1c1 | $1$ | $ 5 \cdot 19 \cdot 61 $ | $x^{2} - x + 1449$ | $C_2$ (as 2T1) | $1$ | $-1$ |
| *2 | 2.5_19_61.4t3.17c1 | $2$ | $ 5 \cdot 19 \cdot 61 $ | $x^{8} - 4 x^{7} + 24 x^{6} - 58 x^{5} + 811 x^{4} - 1530 x^{3} + 12626 x^{2} - 11870 x + 115355$ | $D_4$ (as 8T4) | $1$ | $0$ |