Normalized defining polynomial
\( x^{8} - 2x^{7} + 3x^{6} - 4x^{5} + 14x^{4} - 13x^{3} + 12x^{2} - 11x + 2 \)
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: | \(-4565326108\) \(\medspace = -\,2^{2}\cdot 7\cdot 113^{4}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(16.12\) | 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: | $2^{2/3}7^{1/2}113^{1/2}\approx 44.64521363969143$ | ||
Ramified primes: | \(2\), \(7\), \(113\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{-7}) \) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{233}a^{7}-\frac{104}{233}a^{6}-\frac{107}{233}a^{5}-\frac{41}{233}a^{4}+\frac{2}{233}a^{3}+\frac{16}{233}a^{2}+\frac{11}{233}a+\frac{32}{233}$
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: | $\frac{77}{233}a^{7}-\frac{86}{233}a^{6}+\frac{149}{233}a^{5}-\frac{128}{233}a^{4}+\frac{853}{233}a^{3}-\frac{166}{233}a^{2}+\frac{847}{233}a-\frac{99}{233}$, $\frac{52}{233}a^{7}-\frac{49}{233}a^{6}+\frac{28}{233}a^{5}-\frac{35}{233}a^{4}+\frac{570}{233}a^{3}-\frac{100}{233}a^{2}+\frac{106}{233}a-\frac{433}{233}$, $\frac{96}{233}a^{7}-\frac{198}{233}a^{6}+\frac{213}{233}a^{5}-\frac{208}{233}a^{4}+\frac{1124}{233}a^{3}-\frac{1027}{233}a^{2}+\frac{124}{233}a+\frac{43}{233}$, $\frac{315}{233}a^{7}-\frac{606}{233}a^{6}+\frac{1012}{233}a^{5}-\frac{1265}{233}a^{4}+\frac{4358}{233}a^{3}-\frac{3814}{233}a^{2}+\frac{4630}{233}a-\frac{2735}{233}$ | 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: | \( 159.944911918 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{3}\cdot 159.944911918 \cdot 1}{2\cdot\sqrt{4565326108}}\cr\approx \mathstrut & 1.17436777941 \end{aligned}\]
Galois group
$S_4\wr C_2$ (as 8T47):
A solvable group of order 1152 |
The 20 conjugacy class representatives for $S_4\wr C_2$ |
Character table for $S_4\wr C_2$ |
Intermediate fields
\(\Q(\sqrt{113}) \) |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 12 siblings: | data not computed |
Degree 16 siblings: | data not computed |
Degree 18 siblings: | data not computed |
Degree 24 siblings: | data not computed |
Degree 32 siblings: | data not computed |
Degree 36 siblings: | data not computed |
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 | ${\href{/padicField/3.8.0.1}{8} }$ | ${\href{/padicField/5.8.0.1}{8} }$ | R | ${\href{/padicField/11.3.0.1}{3} }^{2}{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ | ${\href{/padicField/13.4.0.1}{4} }{,}\,{\href{/padicField/13.3.0.1}{3} }{,}\,{\href{/padicField/13.1.0.1}{1} }$ | ${\href{/padicField/17.8.0.1}{8} }$ | ${\href{/padicField/19.4.0.1}{4} }{,}\,{\href{/padicField/19.2.0.1}{2} }^{2}$ | ${\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.2.0.1}{2} }$ | ${\href{/padicField/29.4.0.1}{4} }^{2}$ | ${\href{/padicField/31.3.0.1}{3} }{,}\,{\href{/padicField/31.2.0.1}{2} }{,}\,{\href{/padicField/31.1.0.1}{1} }^{3}$ | ${\href{/padicField/37.6.0.1}{6} }{,}\,{\href{/padicField/37.2.0.1}{2} }$ | ${\href{/padicField/41.3.0.1}{3} }{,}\,{\href{/padicField/41.2.0.1}{2} }{,}\,{\href{/padicField/41.1.0.1}{1} }^{3}$ | ${\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.2.0.1}{2} }$ | ${\href{/padicField/47.4.0.1}{4} }{,}\,{\href{/padicField/47.2.0.1}{2} }^{2}$ | ${\href{/padicField/53.3.0.1}{3} }{,}\,{\href{/padicField/53.2.0.1}{2} }^{2}{,}\,{\href{/padicField/53.1.0.1}{1} }$ | ${\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.2.0.1}{2} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | $\Q_{2}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
2.3.2.1 | $x^{3} + 2$ | $3$ | $1$ | $2$ | $S_3$ | $[\ ]_{3}^{2}$ | |
2.4.0.1 | $x^{4} + x + 1$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
\(7\) | $\Q_{7}$ | $x + 4$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
7.2.1.1 | $x^{2} + 21$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
7.2.0.1 | $x^{2} + 6 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
7.3.0.1 | $x^{3} + 6 x^{2} + 4$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
\(113\) | 113.2.1.1 | $x^{2} + 113$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
113.6.3.1 | $x^{6} + 28137 x^{5} + 263897278 x^{4} + 825036406249 x^{3} + 31934661124 x^{2} + 6629626445831 x + 90832452871289$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
1.7.2t1.a.a | $1$ | $ 7 $ | \(\Q(\sqrt{-7}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
1.791.2t1.a.a | $1$ | $ 7 \cdot 113 $ | \(\Q(\sqrt{-791}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
* | 1.113.2t1.a.a | $1$ | $ 113 $ | \(\Q(\sqrt{113}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
2.791.4t3.c.a | $2$ | $ 7 \cdot 113 $ | 4.0.5537.1 | $D_{4}$ (as 4T3) | $1$ | $0$ | |
4.1131231248.12t34.c.a | $4$ | $ 2^{4} \cdot 7^{2} \cdot 113^{3}$ | 6.0.620144.1 | $C_3^2:D_4$ (as 6T13) | $1$ | $0$ | |
4.17519068.12t34.c.a | $4$ | $ 2^{2} \cdot 7^{3} \cdot 113^{2}$ | 6.0.620144.1 | $C_3^2:D_4$ (as 6T13) | $1$ | $-2$ | |
4.88592.6t13.a.a | $4$ | $ 2^{4} \cdot 7^{2} \cdot 113 $ | 6.0.620144.1 | $C_3^2:D_4$ (as 6T13) | $1$ | $0$ | |
4.357532.6t13.a.a | $4$ | $ 2^{2} \cdot 7 \cdot 113^{2}$ | 6.0.620144.1 | $C_3^2:D_4$ (as 6T13) | $1$ | $2$ | |
6.13857582788.12t201.a.a | $6$ | $ 2^{2} \cdot 7^{4} \cdot 113^{3}$ | 8.2.4565326108.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $2$ | |
6.97003079516.12t202.a.a | $6$ | $ 2^{2} \cdot 7^{5} \cdot 113^{3}$ | 8.2.4565326108.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $0$ | |
* | 6.40401116.8t47.a.a | $6$ | $ 2^{2} \cdot 7 \cdot 113^{3}$ | 8.2.4565326108.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $0$ |
6.282807812.12t200.a.a | $6$ | $ 2^{2} \cdot 7^{2} \cdot 113^{3}$ | 8.2.4565326108.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $-2$ | |
9.31674474944.16t1294.a.a | $9$ | $ 2^{6} \cdot 7^{3} \cdot 113^{3}$ | 8.2.4565326108.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $-1$ | |
9.108...792.18t272.a.a | $9$ | $ 2^{6} \cdot 7^{6} \cdot 113^{3}$ | 8.2.4565326108.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $1$ | |
9.156...424.18t273.a.a | $9$ | $ 2^{6} \cdot 7^{6} \cdot 113^{6}$ | 8.2.4565326108.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $1$ | |
9.457...768.18t274.a.a | $9$ | $ 2^{6} \cdot 7^{3} \cdot 113^{6}$ | 8.2.4565326108.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $-1$ | |
12.175...488.36t1763.a.a | $12$ | $ 2^{10} \cdot 7^{7} \cdot 113^{6}$ | 8.2.4565326108.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $-2$ | |
12.358...112.24t2821.a.a | $12$ | $ 2^{10} \cdot 7^{5} \cdot 113^{6}$ | 8.2.4565326108.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $2$ | |
18.496...256.36t1758.a.a | $18$ | $ 2^{12} \cdot 7^{9} \cdot 113^{9}$ | 8.2.4565326108.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $0$ |