Normalized defining polynomial
\( x^{8} + 8x^{6} + 14x^{4} - 24x^{2} - 59 \)
Invariants
Degree: | $8$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[2, 3]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(-13459718144\) \(\medspace = -\,2^{16}\cdot 59^{3}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(18.46\) | sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
magma: Abs(Discriminant(OK))^(1/Degree(K));
oscar: (1.0 * dK)^(1/degree(K))
| |
Galois root discriminant: | $2^{2}59^{1/2}\approx 30.72458299147443$ | ||
Ramified primes: | \(2\), \(59\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{-59}) \) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
This field is not Galois over $\Q$. | |||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{4}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a+\frac{1}{4}$, $\frac{1}{4}a^{5}-\frac{1}{2}a^{2}+\frac{1}{4}a-\frac{1}{2}$, $\frac{1}{4}a^{6}-\frac{1}{2}a^{3}+\frac{1}{4}a^{2}-\frac{1}{2}a$, $\frac{1}{4}a^{7}+\frac{1}{4}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$
Unit group
Rank: | $4$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
Torsion generator: | \( -1 \) (order $2$) | sage: UK.torsion_generator()
gp: K.tu[2]
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
oscar: torsion_units_generator(OK)
| |
Fundamental units: | $\frac{1}{4}a^{6}+\frac{3}{4}a^{4}-\frac{3}{4}a^{2}-\frac{9}{4}$, $\frac{1}{4}a^{6}+\frac{5}{4}a^{4}+\frac{1}{4}a^{2}-\frac{11}{4}$, $\frac{1}{4}a^{6}+\frac{1}{4}a^{5}+\frac{3}{2}a^{4}+\frac{3}{2}a^{3}+\frac{7}{4}a^{2}+\frac{11}{4}a-1$, $\frac{3}{4}a^{7}+\frac{13}{4}a^{5}-\frac{5}{4}a^{4}-\frac{7}{4}a^{3}-a^{2}-\frac{49}{4}a+\frac{27}{4}$ | sage: UK.fundamental_units()
gp: K.fu
magma: [K|fUK(g): g in Generators(UK)];
oscar: [K(fUK(a)) for a in gens(UK)]
| |
Regulator: | \( 303.921985399 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{3}\cdot 303.921985399 \cdot 1}{2\cdot\sqrt{13459718144}}\cr\approx \mathstrut & 1.29961206831 \end{aligned}\]
Galois group
$\GL(2,3)$ (as 8T23):
A solvable group of order 48 |
The 8 conjugacy class representatives for $\textrm{GL(2,3)}$ |
Character table for $\textrm{GL(2,3)}$ |
Intermediate fields
4.2.3776.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 16 sibling: | deg 16 |
Degree 24 sibling: | deg 24 |
Arithmetically equvalently sibling: | 8.2.13459718144.4 |
Minimal sibling: | This field is its own minimal sibling |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | ${\href{/padicField/3.3.0.1}{3} }^{2}{,}\,{\href{/padicField/3.1.0.1}{1} }^{2}$ | ${\href{/padicField/5.3.0.1}{3} }^{2}{,}\,{\href{/padicField/5.1.0.1}{1} }^{2}$ | ${\href{/padicField/7.6.0.1}{6} }{,}\,{\href{/padicField/7.2.0.1}{2} }$ | ${\href{/padicField/11.8.0.1}{8} }$ | ${\href{/padicField/13.8.0.1}{8} }$ | ${\href{/padicField/17.4.0.1}{4} }^{2}$ | ${\href{/padicField/19.3.0.1}{3} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | ${\href{/padicField/23.8.0.1}{8} }$ | ${\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.2.0.1}{2} }$ | ${\href{/padicField/31.2.0.1}{2} }^{3}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.8.0.1}{8} }$ | ${\href{/padicField/41.3.0.1}{3} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.2.0.1}{2} }^{3}{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | ${\href{/padicField/47.8.0.1}{8} }$ | ${\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.2.0.1}{2} }$ | R |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.8.16.66 | $x^{8} + 2 x^{6} + 4 x^{3} + 4 x + 6$ | $8$ | $1$ | $16$ | $QD_{16}$ | $[2, 2, 5/2]^{2}$ |
\(59\) | 59.2.1.1 | $x^{2} + 118$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
59.2.0.1 | $x^{2} + 58 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
59.4.2.1 | $x^{4} + 116 x^{3} + 3486 x^{2} + 7076 x + 201725$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
1.59.2t1.a.a | $1$ | $ 59 $ | \(\Q(\sqrt{-59}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
2.59.3t2.a.a | $2$ | $ 59 $ | 3.1.59.1 | $S_3$ (as 3T2) | $1$ | $0$ | |
2.1888.24t22.a.a | $2$ | $ 2^{5} \cdot 59 $ | 8.2.13459718144.1 | $\textrm{GL(2,3)}$ (as 8T23) | $0$ | $0$ | |
2.1888.24t22.a.b | $2$ | $ 2^{5} \cdot 59 $ | 8.2.13459718144.1 | $\textrm{GL(2,3)}$ (as 8T23) | $0$ | $0$ | |
3.222784.6t8.f.a | $3$ | $ 2^{6} \cdot 59^{2}$ | 4.2.3776.1 | $S_4$ (as 4T5) | $1$ | $-1$ | |
* | 3.3776.4t5.c.a | $3$ | $ 2^{6} \cdot 59 $ | 4.2.3776.1 | $S_4$ (as 4T5) | $1$ | $1$ |
* | 4.3564544.8t23.a.a | $4$ | $ 2^{10} \cdot 59^{2}$ | 8.2.13459718144.1 | $\textrm{GL(2,3)}$ (as 8T23) | $1$ | $0$ |