Normalized defining polynomial
\( x^{8} - 20x^{6} + 42x^{4} - 24x^{2} + 4 \)
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)
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Discriminant: | \(3873527824384\) \(\medspace = 2^{22}\cdot 31^{4}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(37.46\) | 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: | $2^{11/4}31^{2/3}\approx 66.38555910891195$ | ||
Ramified primes: | \(2\), \(31\) | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2}a^{4}$, $\frac{1}{2}a^{5}$, $\frac{1}{2}a^{6}$, $\frac{1}{2}a^{7}$
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: | $a^{6}-\frac{39}{2}a^{4}+32a^{2}-8$, $\frac{3}{2}a^{6}-\frac{59}{2}a^{4}+53a^{2}-18$, $3a^{7}-\frac{3}{2}a^{6}-\frac{117}{2}a^{5}+\frac{59}{2}a^{4}+97a^{3}-53a^{2}-28a+15$, $\frac{1}{2}a^{7}+\frac{3}{2}a^{6}-10a^{5}-\frac{59}{2}a^{4}+21a^{3}+53a^{2}-12a-16$, $\frac{11}{2}a^{7}-\frac{7}{2}a^{6}-108a^{5}+\frac{137}{2}a^{4}+192a^{3}-118a^{2}-67a+40$, $2a^{7}+\frac{5}{2}a^{6}-\frac{77}{2}a^{5}-\frac{97}{2}a^{4}+55a^{3}+76a^{2}-4a-15$, $2a^{7}-\frac{5}{2}a^{6}-\frac{77}{2}a^{5}+\frac{97}{2}a^{4}+55a^{3}-76a^{2}-4a+15$ | sage: UK.fundamental_units()
gp: K.fu
magma: [K|fUK(g): g in Generators(UK)];
oscar: [K(fUK(a)) for a in gens(UK)]
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Regulator: | \( 11513.8676242 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{8}\cdot(2\pi)^{0}\cdot 11513.8676242 \cdot 1}{2\cdot\sqrt{3873527824384}}\cr\approx \mathstrut & 0.748820735185 \end{aligned}\]
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
4.4.61504.1 |
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 | R | ${\href{/padicField/3.3.0.1}{3} }^{2}{,}\,{\href{/padicField/3.1.0.1}{1} }^{2}$ | ${\href{/padicField/5.6.0.1}{6} }{,}\,{\href{/padicField/5.2.0.1}{2} }$ | ${\href{/padicField/7.6.0.1}{6} }{,}\,{\href{/padicField/7.2.0.1}{2} }$ | ${\href{/padicField/11.3.0.1}{3} }^{2}{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ | ${\href{/padicField/13.3.0.1}{3} }^{2}{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ | ${\href{/padicField/17.6.0.1}{6} }{,}\,{\href{/padicField/17.2.0.1}{2} }$ | ${\href{/padicField/19.3.0.1}{3} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | ${\href{/padicField/23.4.0.1}{4} }^{2}$ | ${\href{/padicField/29.4.0.1}{4} }^{2}$ | R | ${\href{/padicField/37.3.0.1}{3} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.3.0.1}{3} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.3.0.1}{3} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | ${\href{/padicField/47.4.0.1}{4} }^{2}$ | ${\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.2.0.1}{2} }$ | ${\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.2.0.1}{2} }$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.8.22.4 | $x^{8} + 16 x^{7} + 68 x^{6} + 24 x^{5} + 48 x^{4} + 128 x^{3} + 568 x^{2} + 336 x + 588$ | $4$ | $2$ | $22$ | $Q_8$ | $[3, 4]^{2}$ |
\(31\) | 31.2.0.1 | $x^{2} + 29 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
31.6.4.1 | $x^{6} + 87 x^{5} + 2532 x^{4} + 24973 x^{3} + 10293 x^{2} + 78438 x + 748956$ | $3$ | $2$ | $4$ | $C_6$ | $[\ ]_{3}^{2}$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
1.31.3t1.a.a | $1$ | $ 31 $ | 3.3.961.1 | $C_3$ (as 3T1) | $0$ | $1$ | |
1.31.3t1.a.b | $1$ | $ 31 $ | 3.3.961.1 | $C_3$ (as 3T1) | $0$ | $1$ | |
2.246016.24t7.f.a | $2$ | $ 2^{8} \cdot 31^{2}$ | 8.8.3873527824384.2 | $\SL(2,3)$ (as 8T12) | $-1$ | $2$ | |
* | 2.7936.8t12.b.a | $2$ | $ 2^{8} \cdot 31 $ | 8.8.3873527824384.2 | $\SL(2,3)$ (as 8T12) | $0$ | $2$ |
* | 2.7936.8t12.b.b | $2$ | $ 2^{8} \cdot 31 $ | 8.8.3873527824384.2 | $\SL(2,3)$ (as 8T12) | $0$ | $2$ |
* | 3.61504.4t4.a.a | $3$ | $ 2^{6} \cdot 31^{2}$ | 4.4.61504.1 | $A_4$ (as 4T4) | $1$ | $3$ |