Normalized defining polynomial
\( x^{8} - 19x^{6} + 79x^{4} - 98x^{2} + 9 \)
Invariants
Degree: | $8$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[8, 0]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(338920744561\) \(\medspace = 7^{4}\cdot 109^{4}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(27.62\) | 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: | $7^{2/3}109^{2/3}\approx 83.49933665075943$ | ||
Ramified primes: | \(7\), \(109\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q\) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{2}a^{5}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{6}a^{7}-\frac{1}{6}a^{5}-\frac{1}{3}a^{3}-\frac{1}{2}a^{2}+\frac{1}{6}a$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$
Unit group
Rank: | $7$ | 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: | $\frac{5}{6}a^{7}+a^{6}-\frac{83}{6}a^{5}-\frac{33}{2}a^{4}+\frac{197}{6}a^{3}+38a^{2}-\frac{37}{6}a-\frac{9}{2}$, $\frac{7}{6}a^{7}-2a^{6}-\frac{115}{6}a^{5}+33a^{4}+\frac{128}{3}a^{3}-\frac{151}{2}a^{2}-\frac{11}{6}a+8$, $\frac{7}{6}a^{7}+2a^{6}-\frac{115}{6}a^{5}-33a^{4}+\frac{128}{3}a^{3}+\frac{151}{2}a^{2}-\frac{11}{6}a-8$, $\frac{4}{3}a^{7}-\frac{5}{2}a^{6}-\frac{131}{6}a^{5}+41a^{4}+\frac{287}{6}a^{3}-91a^{2}-\frac{8}{3}a+\frac{15}{2}$, $\frac{7}{6}a^{7}-\frac{115}{6}a^{5}+\frac{128}{3}a^{3}+\frac{1}{2}a^{2}-\frac{17}{6}a$, $\frac{13}{6}a^{7}-\frac{107}{3}a^{5}+\frac{242}{3}a^{3}-\frac{25}{3}a+\frac{1}{2}$, $a^{7}-\frac{1}{2}a^{6}-17a^{5}+\frac{17}{2}a^{4}+46a^{3}-23a^{2}-\frac{45}{2}a+\frac{23}{2}$ | 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: | \( 2009.54817518 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{8}\cdot(2\pi)^{0}\cdot 2009.54817518 \cdot 1}{2\cdot\sqrt{338920744561}}\cr\approx \mathstrut & 0.441834186333 \end{aligned}\]
Galois group
$\SL(2,3)$ (as 8T12):
A solvable group of order 24 |
The 7 conjugacy class representatives for $\SL(2,3)$ |
Character table for $\SL(2,3)$ |
Intermediate fields
4.4.582169.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Galois closure: | deg 24 |
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 | ${\href{/padicField/2.6.0.1}{6} }{,}\,{\href{/padicField/2.2.0.1}{2} }$ | ${\href{/padicField/3.6.0.1}{6} }{,}\,{\href{/padicField/3.2.0.1}{2} }$ | ${\href{/padicField/5.4.0.1}{4} }^{2}$ | R | ${\href{/padicField/11.4.0.1}{4} }^{2}$ | ${\href{/padicField/13.6.0.1}{6} }{,}\,{\href{/padicField/13.2.0.1}{2} }$ | ${\href{/padicField/17.6.0.1}{6} }{,}\,{\href{/padicField/17.2.0.1}{2} }$ | ${\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.2.0.1}{2} }$ | ${\href{/padicField/23.3.0.1}{3} }^{2}{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ | ${\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.2.0.1}{2} }$ | ${\href{/padicField/31.4.0.1}{4} }^{2}$ | ${\href{/padicField/37.3.0.1}{3} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.4.0.1}{4} }^{2}$ | ${\href{/padicField/43.4.0.1}{4} }^{2}$ | ${\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.2.0.1}{2} }$ | ${\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.2.0.1}{2} }$ | ${\href{/padicField/59.4.0.1}{4} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(7\) | $\Q_{7}$ | $x + 4$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{7}$ | $x + 4$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
7.3.2.3 | $x^{3} + 21$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ | |
7.3.2.3 | $x^{3} + 21$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ | |
\(109\) | 109.2.0.1 | $x^{2} + 108 x + 6$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
109.6.4.3 | $x^{6} + 10682 x^{3} - 6047429$ | $3$ | $2$ | $4$ | $C_6$ | $[\ ]_{3}^{2}$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
1.763.3t1.b.a | $1$ | $ 7 \cdot 109 $ | 3.3.582169.2 | $C_3$ (as 3T1) | $0$ | $1$ | |
1.763.3t1.b.b | $1$ | $ 7 \cdot 109 $ | 3.3.582169.2 | $C_3$ (as 3T1) | $0$ | $1$ | |
2.582169.24t7.a.a | $2$ | $ 7^{2} \cdot 109^{2}$ | 8.8.338920744561.1 | $\SL(2,3)$ (as 8T12) | $-1$ | $2$ | |
* | 2.763.8t12.a.a | $2$ | $ 7 \cdot 109 $ | 8.8.338920744561.1 | $\SL(2,3)$ (as 8T12) | $0$ | $2$ |
* | 2.763.8t12.a.b | $2$ | $ 7 \cdot 109 $ | 8.8.338920744561.1 | $\SL(2,3)$ (as 8T12) | $0$ | $2$ |
* | 3.582169.4t4.a.a | $3$ | $ 7^{2} \cdot 109^{2}$ | 4.4.582169.1 | $A_4$ (as 4T4) | $1$ | $3$ |