Normalized defining polynomial
\( x^{8} - 4 x^{7} - 24 x^{6} + 86 x^{5} - 149 x^{4} + 150 x^{3} + 11642 x^{2} - 11702 x + 22163 \)
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: | \(3042661895388241=7^{4}\cdot 1061^{4}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $86.18$ | 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: | $7, 1061$ | 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 \(a\))
$1$, $a$, $\frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{4} a^{4} - \frac{1}{4} a^{2} - \frac{1}{2} a - \frac{1}{4}$, $\frac{1}{4} a^{5} - \frac{1}{4} a^{3} + \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{24080} a^{6} - \frac{3}{24080} a^{5} + \frac{171}{1505} a^{4} - \frac{781}{3440} a^{3} + \frac{87}{430} a^{2} - \frac{2139}{24080} a - \frac{1921}{24080}$, $\frac{1}{7633360} a^{7} + \frac{31}{1526672} a^{6} + \frac{467681}{3816680} a^{5} - \frac{951759}{7633360} a^{4} + \frac{59479}{545240} a^{3} - \frac{1345383}{7633360} a^{2} + \frac{1977817}{7633360} a + \frac{916791}{3816680}$
Class group and class number
$C_{3}\times C_{21}$, which has order $63$
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: | \( 1180.01058514 \) | 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
| \(\Q(\sqrt{-7427}) \), \(\Q(\sqrt{-7}) \), \(\Q(\sqrt{1061}) \), \(\Q(\sqrt{-7}, \sqrt{1061})\), 4.0.51989.1 x2, 4.2.7880047.1 x2 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Degree 4 siblings: | 4.2.7880047.1, 4.0.51989.1 |
Frobenius cycle types
| $p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 | 53 | 59 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/LocalNumberField/2.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/3.4.0.1}{4} }^{2}$ | ${\href{/LocalNumberField/5.2.0.1}{2} }^{4}$ | R | ${\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}$ | ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}$ | ${\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/23.1.0.1}{1} }^{8}$ | ${\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/37.1.0.1}{1} }^{8}$ | ${\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/53.1.0.1}{1} }^{8}$ | ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
| $7$ | 7.4.2.1 | $x^{4} + 35 x^{2} + 441$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
| 7.4.2.1 | $x^{4} + 35 x^{2} + 441$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 1061 | Data not computed | ||||||
Artin representations
| Label | Dimension | Conductor | Defining polynomial of Artin field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| * | 1.1.1t1.1c1 | $1$ | $1$ | $x$ | $C_1$ | $1$ | $1$ |
| * | 1.1061.2t1.1c1 | $1$ | $ 1061 $ | $x^{2} - x - 265$ | $C_2$ (as 2T1) | $1$ | $1$ |
| * | 1.7.2t1.1c1 | $1$ | $ 7 $ | $x^{2} - x + 2$ | $C_2$ (as 2T1) | $1$ | $-1$ |
| * | 1.7_1061.2t1.1c1 | $1$ | $ 7 \cdot 1061 $ | $x^{2} - x + 1857$ | $C_2$ (as 2T1) | $1$ | $-1$ |
| *2 | 2.7_1061.4t3.3c1 | $2$ | $ 7 \cdot 1061 $ | $x^{8} - 4 x^{7} - 24 x^{6} + 86 x^{5} - 149 x^{4} + 150 x^{3} + 11642 x^{2} - 11702 x + 22163$ | $D_4$ (as 8T4) | $1$ | $0$ |