Normalized defining polynomial
\( x^{8} - x^{7} - 4x^{6} + 4x^{5} + 4x^{4} - 9x^{3} + 2x^{2} - 3x - 5 \)
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: | \(-1327373299\) \(\medspace = -\,7^{3}\cdot 157^{3}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(13.82\) | 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: | $7^{1/2}157^{1/2}\approx 33.15116890850155$ | ||
Ramified primes: | \(7\), \(157\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{-1099}) \) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{337}a^{7}+\frac{72}{337}a^{6}-\frac{140}{337}a^{5}-\frac{106}{337}a^{4}+\frac{17}{337}a^{3}-\frac{116}{337}a^{2}-\frac{41}{337}a+\frac{37}{337}$
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: | $\frac{37}{337}a^{7}-\frac{32}{337}a^{6}-\frac{125}{337}a^{5}+\frac{122}{337}a^{4}-\frac{45}{337}a^{3}-\frac{248}{337}a^{2}+\frac{168}{337}a-\frac{316}{337}$, $\frac{59}{337}a^{7}-\frac{133}{337}a^{6}-\frac{172}{337}a^{5}+\frac{486}{337}a^{4}-\frac{8}{337}a^{3}-\frac{778}{337}a^{2}+\frac{614}{337}a-\frac{176}{337}$, $\frac{22}{337}a^{7}-\frac{101}{337}a^{6}-\frac{47}{337}a^{5}+\frac{364}{337}a^{4}+\frac{37}{337}a^{3}-\frac{193}{337}a^{2}+\frac{446}{337}a-\frac{534}{337}$, $\frac{18}{337}a^{7}-\frac{52}{337}a^{6}-\frac{161}{337}a^{5}+\frac{114}{337}a^{4}+\frac{306}{337}a^{3}-\frac{66}{337}a^{2}-\frac{64}{337}a-\frac{8}{337}$ | 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: | \( 28.7258633658 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{3}\cdot 28.7258633658 \cdot 1}{2\cdot\sqrt{1327373299}}\cr\approx \mathstrut & 0.391152538011 \end{aligned}\]
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
4.2.1099.1 |
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: | 8.2.1327373299.4 |
Minimal sibling: | 8.2.1327373299.4 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | ${\href{/padicField/2.8.0.1}{8} }$ | ${\href{/padicField/3.8.0.1}{8} }$ | ${\href{/padicField/5.3.0.1}{3} }^{2}{,}\,{\href{/padicField/5.1.0.1}{1} }^{2}$ | R | ${\href{/padicField/11.3.0.1}{3} }^{2}{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ | ${\href{/padicField/13.8.0.1}{8} }$ | ${\href{/padicField/17.8.0.1}{8} }$ | ${\href{/padicField/19.8.0.1}{8} }$ | ${\href{/padicField/23.8.0.1}{8} }$ | ${\href{/padicField/29.2.0.1}{2} }^{3}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.8.0.1}{8} }$ | ${\href{/padicField/37.3.0.1}{3} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.4.0.1}{4} }^{2}$ | ${\href{/padicField/43.8.0.1}{8} }$ | ${\href{/padicField/47.2.0.1}{2} }^{3}{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | ${\href{/padicField/53.2.0.1}{2} }^{3}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | ${\href{/padicField/59.3.0.1}{3} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(7\) | 7.2.0.1 | $x^{2} + 6 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
7.2.1.2 | $x^{2} + 7$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
7.4.2.1 | $x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
\(157\) | $\Q_{157}$ | $x + 152$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{157}$ | $x + 152$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
157.2.1.2 | $x^{2} + 314$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
157.2.1.2 | $x^{2} + 314$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
157.2.1.2 | $x^{2} + 314$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
1.1099.2t1.a.a | $1$ | $ 7 \cdot 157 $ | \(\Q(\sqrt{-1099}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
2.1099.3t2.a.a | $2$ | $ 7 \cdot 157 $ | 3.1.1099.1 | $S_3$ (as 3T2) | $1$ | $0$ | |
2.1099.24t22.d.a | $2$ | $ 7 \cdot 157 $ | 8.2.1327373299.5 | $\textrm{GL(2,3)}$ (as 8T23) | $0$ | $0$ | |
2.1099.24t22.d.b | $2$ | $ 7 \cdot 157 $ | 8.2.1327373299.5 | $\textrm{GL(2,3)}$ (as 8T23) | $0$ | $0$ | |
3.1207801.6t8.a.a | $3$ | $ 7^{2} \cdot 157^{2}$ | 4.2.1099.1 | $S_4$ (as 4T5) | $1$ | $-1$ | |
* | 3.1099.4t5.a.a | $3$ | $ 7 \cdot 157 $ | 4.2.1099.1 | $S_4$ (as 4T5) | $1$ | $1$ |
* | 4.1207801.8t23.d.a | $4$ | $ 7^{2} \cdot 157^{2}$ | 8.2.1327373299.5 | $\textrm{GL(2,3)}$ (as 8T23) | $1$ | $0$ |