Normalized defining polynomial
\( x^{8} - 2 x^{7} - 38 x^{6} + 2 x^{5} + 458 x^{4} + 562 x^{3} - 1438 x^{2} - 3402 x - 1899 \)
Invariants
Degree: | $8$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
Signature: | $[8, 0]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
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Discriminant: | \(333621760000=2^{12}\cdot 5^{4}\cdot 19^{4}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
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Root discriminant: | $27.57$ | magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
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Ramified primes: | $2, 5, 19$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
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This field is Galois over $\Q$. | |||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{6} a^{5} + \frac{1}{6} a^{4} - \frac{1}{3} a^{2} - \frac{1}{6} a - \frac{1}{2}$, $\frac{1}{114} a^{6} - \frac{3}{38} a^{5} + \frac{4}{57} a^{4} + \frac{2}{57} a^{3} - \frac{29}{114} a^{2} + \frac{13}{114} a - \frac{8}{19}$, $\frac{1}{114} a^{7} + \frac{1}{38} a^{5} - \frac{1}{6} a^{4} + \frac{7}{114} a^{3} + \frac{28}{57} a^{2} - \frac{7}{114} a - \frac{11}{38}$
Class group and class number
Trivial group, which has order $1$
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]
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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
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Regulator: | \( 1758.1037263 \) | 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
\(\Q(\sqrt{95}) \), \(\Q(\sqrt{19}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{5}, \sqrt{19})\), 4.4.115520.1 x2, 4.4.7600.1 x2 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 4 siblings: | 4.4.115520.1, 4.4.7600.1 |
Frobenius cycle types
$p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 | 53 | 59 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | ${\href{/LocalNumberField/3.2.0.1}{2} }^{4}$ | R | ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}$ | ${\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}$ | ${\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ | R | ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}$ | ${\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}$ | ${\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}$ | ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}$ | ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}$ | ${\href{/LocalNumberField/59.2.0.1}{2} }^{4}$ |
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 |
---|---|---|---|---|---|---|---|
$2$ | 2.8.12.14 | $x^{8} + 12 x^{4} + 144$ | $4$ | $2$ | $12$ | $D_4$ | $[2, 2]^{2}$ |
$5$ | 5.4.2.1 | $x^{4} + 15 x^{2} + 100$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
5.4.2.1 | $x^{4} + 15 x^{2} + 100$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
$19$ | 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 | $x^{2} - 19$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
Artin representations
Label | Dimension | Conductor | Defining polynomial of Artin field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.1c1 | $1$ | $1$ | $x$ | $C_1$ | $1$ | $1$ |
* | 1.5.2t1.1c1 | $1$ | $ 5 $ | $x^{2} - x - 1$ | $C_2$ (as 2T1) | $1$ | $1$ |
* | 1.2e2_19.2t1.1c1 | $1$ | $ 2^{2} \cdot 19 $ | $x^{2} - 19$ | $C_2$ (as 2T1) | $1$ | $1$ |
* | 1.2e2_5_19.2t1.1c1 | $1$ | $ 2^{2} \cdot 5 \cdot 19 $ | $x^{2} - 95$ | $C_2$ (as 2T1) | $1$ | $1$ |
*2 | 2.2e4_5_19.4t3.7c1 | $2$ | $ 2^{4} \cdot 5 \cdot 19 $ | $x^{8} - 2 x^{7} - 38 x^{6} + 2 x^{5} + 458 x^{4} + 562 x^{3} - 1438 x^{2} - 3402 x - 1899$ | $D_4$ (as 8T4) | $1$ | $2$ |