Normalized defining polynomial
\( x^{8} - 15x^{6} + 74x^{4} - 131x^{2} + 64 \)
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: | \(3797883801856\) \(\medspace = 2^{8}\cdot 349^{4}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(37.36\) | 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: | $2\cdot 349^{2/3}\approx 99.1395508408829$ | ||
Ramified primes: | \(2\), \(349\) | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{8}a^{7}+\frac{1}{8}a^{5}+\frac{1}{4}a^{3}-\frac{3}{8}a$
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: | $\frac{1}{8}a^{7}-\frac{15}{8}a^{5}+\frac{33}{4}a^{3}-\frac{75}{8}a$, $\frac{1}{8}a^{7}-\frac{7}{8}a^{5}+\frac{1}{4}a^{3}+\frac{21}{8}a$, $\frac{1}{2}a^{7}+a^{6}-\frac{9}{2}a^{5}-9a^{4}+9a^{3}+18a^{2}-\frac{5}{2}a-7$, $\frac{3}{8}a^{7}-a^{6}-\frac{29}{8}a^{5}+9a^{4}+\frac{39}{4}a^{3}-21a^{2}-\frac{49}{8}a+12$, $\frac{1}{4}a^{7}-\frac{15}{4}a^{5}+\frac{33}{2}a^{3}-a^{2}-\frac{71}{4}a+7$, $\frac{1}{4}a^{7}-\frac{11}{4}a^{5}-a^{4}+\frac{17}{2}a^{3}+8a^{2}-\frac{27}{4}a-11$, $\frac{5}{8}a^{7}-a^{6}-\frac{59}{8}a^{5}+12a^{4}+\frac{93}{4}a^{3}-39a^{2}-\frac{79}{8}a+20$ | 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: | \( 14077.7203941 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{8}\cdot(2\pi)^{0}\cdot 14077.7203941 \cdot 1}{2\cdot\sqrt{3797883801856}}\cr\approx \mathstrut & 0.924637426235 \end{aligned}\]
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
4.4.121801.1 |
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 | ${\href{/padicField/3.6.0.1}{6} }{,}\,{\href{/padicField/3.2.0.1}{2} }$ | ${\href{/padicField/5.3.0.1}{3} }^{2}{,}\,{\href{/padicField/5.1.0.1}{1} }^{2}$ | ${\href{/padicField/7.3.0.1}{3} }^{2}{,}\,{\href{/padicField/7.1.0.1}{1} }^{2}$ | ${\href{/padicField/11.4.0.1}{4} }^{2}$ | ${\href{/padicField/13.3.0.1}{3} }^{2}{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ | ${\href{/padicField/17.4.0.1}{4} }^{2}$ | ${\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.2.0.1}{2} }$ | ${\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.2.0.1}{2} }$ | ${\href{/padicField/29.3.0.1}{3} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.4.0.1}{4} }^{2}$ | ${\href{/padicField/37.2.0.1}{2} }^{4}$ | ${\href{/padicField/41.4.0.1}{4} }^{2}$ | ${\href{/padicField/43.3.0.1}{3} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | ${\href{/padicField/47.4.0.1}{4} }^{2}$ | ${\href{/padicField/53.4.0.1}{4} }^{2}$ | ${\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.2.0.1}{2} }$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.2.2.2 | $x^{2} + 2 x + 6$ | $2$ | $1$ | $2$ | $C_2$ | $[2]$ |
2.6.6.5 | $x^{6} + 6 x^{5} + 32 x^{4} + 90 x^{3} + 199 x^{2} + 204 x + 149$ | $2$ | $3$ | $6$ | $C_6$ | $[2]^{3}$ | |
\(349\) | $\Q_{349}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{349}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
Deg $3$ | $3$ | $1$ | $2$ | ||||
Deg $3$ | $3$ | $1$ | $2$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
1.349.3t1.a.a | $1$ | $ 349 $ | 3.3.121801.1 | $C_3$ (as 3T1) | $0$ | $1$ | |
1.349.3t1.a.b | $1$ | $ 349 $ | 3.3.121801.1 | $C_3$ (as 3T1) | $0$ | $1$ | |
2.1948816.24t7.a.a | $2$ | $ 2^{4} \cdot 349^{2}$ | 8.8.3797883801856.1 | $\SL(2,3)$ (as 8T12) | $-1$ | $2$ | |
* | 2.5584.8t12.a.a | $2$ | $ 2^{4} \cdot 349 $ | 8.8.3797883801856.1 | $\SL(2,3)$ (as 8T12) | $0$ | $2$ |
* | 2.5584.8t12.a.b | $2$ | $ 2^{4} \cdot 349 $ | 8.8.3797883801856.1 | $\SL(2,3)$ (as 8T12) | $0$ | $2$ |
* | 3.121801.4t4.a.a | $3$ | $ 349^{2}$ | 4.4.121801.1 | $A_4$ (as 4T4) | $1$ | $3$ |