Normalized defining polynomial
\( x^{8} - 2x^{7} + 5x^{5} - 6x^{4} - 10x^{3} + 21x^{2} + 5x - 13 \)
Invariants
Degree: | $8$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[2, 3]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(-1280239375\) \(\medspace = -\,5^{4}\cdot 127^{3}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(13.75\) | 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: | $5^{2/3}127^{1/2}\approx 32.95200640537238$ | ||
Ramified primes: | \(5\), \(127\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{-127}) \) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{827}a^{7}-\frac{332}{827}a^{6}+\frac{396}{827}a^{5}-\frac{9}{827}a^{4}-\frac{344}{827}a^{3}+\frac{211}{827}a^{2}-\frac{141}{827}a+\frac{223}{827}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$
Unit group
Rank: | $4$ | 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: | $\frac{75}{827}a^{7}-\frac{90}{827}a^{6}-\frac{72}{827}a^{5}+\frac{152}{827}a^{4}-\frac{163}{827}a^{3}-\frac{715}{827}a^{2}+\frac{1003}{827}a+\frac{1012}{827}$, $\frac{250}{827}a^{7}-\frac{300}{827}a^{6}-\frac{240}{827}a^{5}+\frac{1058}{827}a^{4}-\frac{819}{827}a^{3}-\frac{2659}{827}a^{2}+\frac{2792}{827}a+\frac{2822}{827}$, $\frac{325}{827}a^{7}-\frac{390}{827}a^{6}-\frac{312}{827}a^{5}+\frac{1210}{827}a^{4}-\frac{982}{827}a^{3}-\frac{4201}{827}a^{2}+\frac{2968}{827}a+\frac{3834}{827}$, $\frac{190}{827}a^{7}-\frac{228}{827}a^{6}-\frac{17}{827}a^{5}+\frac{771}{827}a^{4}-\frac{854}{827}a^{3}-\frac{1260}{827}a^{2}+\frac{1328}{827}a+\frac{1020}{827}$ | 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: | \( 32.6933534109 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{3}\cdot 32.6933534109 \cdot 1}{2\cdot\sqrt{1280239375}}\cr\approx \mathstrut & 0.453297662748 \end{aligned}\]
Galois group
$\GL(2,3)$ (as 8T23):
A solvable group of order 48 |
The 8 conjugacy class representatives for $\textrm{GL(2,3)}$ |
Character table for $\textrm{GL(2,3)}$ |
Intermediate fields
4.2.3175.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 16 sibling: | deg 16 |
Degree 24 sibling: | deg 24 |
Arithmetically equvalently sibling: | 8.2.1280239375.1 |
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 | ${\href{/padicField/2.6.0.1}{6} }{,}\,{\href{/padicField/2.2.0.1}{2} }$ | ${\href{/padicField/3.8.0.1}{8} }$ | R | ${\href{/padicField/7.8.0.1}{8} }$ | ${\href{/padicField/11.4.0.1}{4} }^{2}$ | ${\href{/padicField/13.3.0.1}{3} }^{2}{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ | ${\href{/padicField/17.4.0.1}{4} }^{2}$ | ${\href{/padicField/19.3.0.1}{3} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | ${\href{/padicField/23.2.0.1}{2} }^{3}{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ | ${\href{/padicField/29.8.0.1}{8} }$ | ${\href{/padicField/31.4.0.1}{4} }^{2}$ | ${\href{/padicField/37.6.0.1}{6} }{,}\,{\href{/padicField/37.2.0.1}{2} }$ | ${\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.2.0.1}{2} }$ | ${\href{/padicField/43.8.0.1}{8} }$ | ${\href{/padicField/47.3.0.1}{3} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | ${\href{/padicField/53.2.0.1}{2} }^{3}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | ${\href{/padicField/59.8.0.1}{8} }$ |
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 |
---|---|---|---|---|---|---|---|
\(5\) | 5.2.0.1 | $x^{2} + 4 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
5.3.2.1 | $x^{3} + 5$ | $3$ | $1$ | $2$ | $S_3$ | $[\ ]_{3}^{2}$ | |
5.3.2.1 | $x^{3} + 5$ | $3$ | $1$ | $2$ | $S_3$ | $[\ ]_{3}^{2}$ | |
\(127\) | 127.2.1.1 | $x^{2} + 381$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
127.2.0.1 | $x^{2} + 126 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
127.4.2.1 | $x^{4} + 252 x^{3} + 16136 x^{2} + 32760 x + 2031628$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
1.127.2t1.a.a | $1$ | $ 127 $ | \(\Q(\sqrt{-127}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
2.3175.3t2.a.a | $2$ | $ 5^{2} \cdot 127 $ | 3.1.3175.1 | $S_3$ (as 3T2) | $1$ | $0$ | |
2.3175.24t22.a.a | $2$ | $ 5^{2} \cdot 127 $ | 8.2.1280239375.2 | $\textrm{GL(2,3)}$ (as 8T23) | $0$ | $0$ | |
2.3175.24t22.a.b | $2$ | $ 5^{2} \cdot 127 $ | 8.2.1280239375.2 | $\textrm{GL(2,3)}$ (as 8T23) | $0$ | $0$ | |
3.403225.6t8.a.a | $3$ | $ 5^{2} \cdot 127^{2}$ | 4.2.3175.1 | $S_4$ (as 4T5) | $1$ | $-1$ | |
* | 3.3175.4t5.a.a | $3$ | $ 5^{2} \cdot 127 $ | 4.2.3175.1 | $S_4$ (as 4T5) | $1$ | $1$ |
* | 4.403225.8t23.a.a | $4$ | $ 5^{2} \cdot 127^{2}$ | 8.2.1280239375.2 | $\textrm{GL(2,3)}$ (as 8T23) | $1$ | $0$ |