Normalized defining polynomial
\( x^{8} - 3x^{7} - 23x^{6} + 53x^{5} + 132x^{4} - 173x^{3} + 29x^{2} + 6x - 1 \)
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: | \(3679587150625\) \(\medspace = 5^{4}\cdot 277^{4}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(37.22\) | 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: | $5^{1/2}277^{2/3}\approx 95.01777319278573$ | ||
Ramified primes: | \(5\), \(277\) | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{1678}a^{7}-\frac{160}{839}a^{6}-\frac{51}{839}a^{5}+\frac{505}{1678}a^{4}-\frac{543}{1678}a^{3}-\frac{37}{1678}a^{2}-\frac{827}{1678}a-\frac{221}{839}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{2}$, which has order $2$
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: | $\frac{1177}{839}a^{7}-\frac{3285}{839}a^{6}-\frac{27764}{839}a^{5}+\frac{56586}{839}a^{4}+\frac{167168}{839}a^{3}-\frac{168560}{839}a^{2}-\frac{139}{839}a+\frac{6658}{839}$, $\frac{953}{1678}a^{7}-\frac{1460}{839}a^{6}-\frac{10848}{839}a^{5}+\frac{51697}{1678}a^{4}+\frac{121839}{1678}a^{3}-\frac{169501}{1678}a^{2}+\frac{40801}{1678}a-\frac{863}{839}$, $\frac{1655}{1678}a^{7}-\frac{2193}{839}a^{6}-\frac{19802}{839}a^{5}+\frac{73963}{1678}a^{4}+\frac{244053}{1678}a^{3}-\frac{202187}{1678}a^{2}-\frac{19573}{1678}a+\frac{3405}{839}$, $\frac{1243}{1678}a^{7}-\frac{1715}{839}a^{6}-\frac{14731}{839}a^{5}+\frac{58873}{1678}a^{4}+\frac{179153}{1678}a^{3}-\frac{173519}{1678}a^{2}-\frac{9415}{1678}a+\frac{6362}{839}$, $a$, $\frac{918}{839}a^{7}-\frac{2627}{839}a^{6}-\frac{21482}{839}a^{5}+\frac{45768}{839}a^{4}+\frac{127420}{839}a^{3}-\frac{142197}{839}a^{2}+\frac{7660}{839}a+\frac{6193}{839}$, $\frac{2003}{1678}a^{7}-\frac{5837}{1678}a^{6}-\frac{23287}{839}a^{5}+\frac{102039}{1678}a^{4}+\frac{137034}{839}a^{3}-\frac{160808}{839}a^{2}+\frac{9501}{839}a+\frac{14921}{1678}$ | 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: | \( 5687.94563401 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{8}\cdot(2\pi)^{0}\cdot 5687.94563401 \cdot 2}{2\cdot\sqrt{3679587150625}}\cr\approx \mathstrut & 0.759094518269 \end{aligned}\]
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
4.4.76729.1 |
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 | ${\href{/padicField/2.4.0.1}{4} }^{2}$ | ${\href{/padicField/3.3.0.1}{3} }^{2}{,}\,{\href{/padicField/3.1.0.1}{1} }^{2}$ | R | ${\href{/padicField/7.3.0.1}{3} }^{2}{,}\,{\href{/padicField/7.1.0.1}{1} }^{2}$ | ${\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.2.0.1}{2} }$ | ${\href{/padicField/13.4.0.1}{4} }^{2}$ | ${\href{/padicField/17.3.0.1}{3} }^{2}{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ | ${\href{/padicField/19.4.0.1}{4} }^{2}$ | ${\href{/padicField/23.3.0.1}{3} }^{2}{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ | ${\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.2.0.1}{2} }$ | ${\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.2.0.1}{2} }$ | ${\href{/padicField/37.4.0.1}{4} }^{2}$ | ${\href{/padicField/41.4.0.1}{4} }^{2}$ | ${\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.2.0.1}{2} }$ | ${\href{/padicField/47.3.0.1}{3} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | ${\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.2.0.1}{2} }$ | ${\href{/padicField/59.4.0.1}{4} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(5\) | 5.2.1.1 | $x^{2} + 5$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
5.6.3.1 | $x^{6} + 60 x^{5} + 1221 x^{4} + 8846 x^{3} + 9864 x^{2} + 29208 x + 29309$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
\(277\) | Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
Deg $6$ | $3$ | $2$ | $4$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
1.277.3t1.a.a | $1$ | $ 277 $ | 3.3.76729.1 | $C_3$ (as 3T1) | $0$ | $1$ | |
1.277.3t1.a.b | $1$ | $ 277 $ | 3.3.76729.1 | $C_3$ (as 3T1) | $0$ | $1$ | |
2.1918225.24t7.a.a | $2$ | $ 5^{2} \cdot 277^{2}$ | 8.8.3679587150625.1 | $\SL(2,3)$ (as 8T12) | $-1$ | $2$ | |
* | 2.6925.8t12.a.a | $2$ | $ 5^{2} \cdot 277 $ | 8.8.3679587150625.1 | $\SL(2,3)$ (as 8T12) | $0$ | $2$ |
* | 2.6925.8t12.a.b | $2$ | $ 5^{2} \cdot 277 $ | 8.8.3679587150625.1 | $\SL(2,3)$ (as 8T12) | $0$ | $2$ |
* | 3.76729.4t4.a.a | $3$ | $ 277^{2}$ | 4.4.76729.1 | $A_4$ (as 4T4) | $1$ | $3$ |