Normalized defining polynomial
\( x^{8} - 4x^{7} + 17x^{6} - 13x^{5} + 73x^{4} + 88x^{3} + 248x^{2} + 304x + 208 \)
Invariants
Degree: | $8$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[0, 4]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(1616465588409\) \(\medspace = 3^{6}\cdot 7^{4}\cdot 31^{4}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(33.58\) | 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: | $3^{3/4}7^{2/3}31^{2/3}\approx 82.31533758230549$ | ||
Ramified primes: | \(3\), \(7\), \(31\) | 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 a CM field. | |||
Reflex fields: | 8.0.1616465588409.2$^{8}$ |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{8}a^{5}+\frac{1}{4}a^{4}+\frac{1}{8}a^{3}+\frac{1}{8}a^{2}+\frac{3}{8}a-\frac{1}{4}$, $\frac{1}{64}a^{6}+\frac{13}{64}a^{4}-\frac{25}{64}a^{3}-\frac{15}{64}a^{2}-\frac{3}{16}$, $\frac{1}{15872}a^{7}-\frac{9}{7936}a^{6}+\frac{269}{15872}a^{5}-\frac{3779}{15872}a^{4}+\frac{173}{512}a^{3}+\frac{2183}{7936}a^{2}+\frac{653}{3968}a-\frac{565}{1984}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{6}$, which has order $6$
Unit group
Rank: | $3$ | 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: | $\frac{9}{512}a^{7}-\frac{25}{256}a^{6}+\frac{181}{512}a^{5}-\frac{171}{512}a^{4}+\frac{219}{512}a^{3}+\frac{279}{256}a^{2}+\frac{101}{128}a-\frac{21}{64}$, $\frac{3}{64}a^{6}-\frac{1}{8}a^{5}+\frac{23}{64}a^{4}+\frac{45}{64}a^{3}+\frac{139}{64}a^{2}+\frac{21}{8}a+\frac{27}{16}$, $\frac{25}{512}a^{7}-\frac{37}{256}a^{6}+\frac{325}{512}a^{5}+\frac{13}{512}a^{4}+\frac{1539}{512}a^{3}+\frac{2219}{256}a^{2}+\frac{1157}{128}a+\frac{1247}{64}$ | 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: | \( 505.296367936 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{4}\cdot 505.296367936 \cdot 6}{2\cdot\sqrt{1616465588409}}\cr\approx \mathstrut & 1.85824799483 \end{aligned}\]
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
4.4.423801.1 |
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 | ${\href{/padicField/2.3.0.1}{3} }^{2}{,}\,{\href{/padicField/2.1.0.1}{1} }^{2}$ | R | ${\href{/padicField/5.4.0.1}{4} }^{2}$ | R | ${\href{/padicField/11.3.0.1}{3} }^{2}{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ | ${\href{/padicField/13.3.0.1}{3} }^{2}{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ | ${\href{/padicField/17.4.0.1}{4} }^{2}$ | ${\href{/padicField/19.3.0.1}{3} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | ${\href{/padicField/23.3.0.1}{3} }^{2}{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ | ${\href{/padicField/29.4.0.1}{4} }^{2}$ | R | ${\href{/padicField/37.3.0.1}{3} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.3.0.1}{3} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.3.0.1}{3} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | ${\href{/padicField/47.3.0.1}{3} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | ${\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.2.0.1}{2} }$ | ${\href{/padicField/59.4.0.1}{4} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(3\) | 3.8.6.1 | $x^{8} + 9$ | $4$ | $2$ | $6$ | $Q_8$ | $[\ ]_{4}^{2}$ |
\(7\) | 7.2.0.1 | $x^{2} + 6 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
7.6.4.2 | $x^{6} - 42 x^{3} + 147$ | $3$ | $2$ | $4$ | $C_6$ | $[\ ]_{3}^{2}$ | |
\(31\) | $\Q_{31}$ | $x + 28$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{31}$ | $x + 28$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
31.3.2.3 | $x^{3} + 155$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ | |
31.3.2.3 | $x^{3} + 155$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
1.217.3t1.b.a | $1$ | $ 7 \cdot 31 $ | 3.3.47089.2 | $C_3$ (as 3T1) | $0$ | $1$ | |
1.217.3t1.b.b | $1$ | $ 7 \cdot 31 $ | 3.3.47089.2 | $C_3$ (as 3T1) | $0$ | $1$ | |
2.423801.24t7.b.a | $2$ | $ 3^{2} \cdot 7^{2} \cdot 31^{2}$ | 8.0.1616465588409.2 | $\SL(2,3)$ (as 8T12) | $-1$ | $-2$ | |
* | 2.1953.8t12.b.a | $2$ | $ 3^{2} \cdot 7 \cdot 31 $ | 8.0.1616465588409.2 | $\SL(2,3)$ (as 8T12) | $0$ | $-2$ |
* | 2.1953.8t12.b.b | $2$ | $ 3^{2} \cdot 7 \cdot 31 $ | 8.0.1616465588409.2 | $\SL(2,3)$ (as 8T12) | $0$ | $-2$ |
* | 3.423801.4t4.a.a | $3$ | $ 3^{2} \cdot 7^{2} \cdot 31^{2}$ | 4.4.423801.1 | $A_4$ (as 4T4) | $1$ | $3$ |