Normalized defining polynomial
\( x^{8} - 2x^{7} - 2x^{6} + 4x^{5} + 4x^{4} + 4x^{3} - 8x^{2} - 32x + 4 \)
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: | \(-19680620544\) \(\medspace = -\,2^{12}\cdot 3^{7}\cdot 13^{3}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(19.35\) | 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: | $2^{19/12}3^{43/36}13^{3/4}\approx 76.20494829057675$ | ||
Ramified primes: | \(2\), \(3\), \(13\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{-39}) \) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2}a^{4}$, $\frac{1}{2}a^{5}$, $\frac{1}{6}a^{6}-\frac{1}{3}a^{3}-\frac{1}{3}$, $\frac{1}{180}a^{7}+\frac{1}{90}a^{6}+\frac{1}{30}a^{5}+\frac{7}{45}a^{4}+\frac{13}{90}a^{3}-\frac{2}{5}a^{2}+\frac{16}{45}a+\frac{11}{45}$
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: | $\frac{2}{45}a^{7}-\frac{7}{90}a^{6}-\frac{7}{30}a^{5}+\frac{11}{45}a^{4}+\frac{22}{45}a^{3}-\frac{1}{5}a^{2}-\frac{7}{45}a+\frac{13}{45}$, $\frac{13}{180}a^{7}-\frac{17}{90}a^{6}-\frac{1}{15}a^{5}+\frac{47}{90}a^{4}-\frac{41}{90}a^{3}+\frac{4}{5}a^{2}-\frac{62}{45}a-\frac{7}{45}$, $\frac{1}{60}a^{7}+\frac{1}{30}a^{6}+\frac{1}{10}a^{5}-\frac{1}{30}a^{4}-\frac{17}{30}a^{3}-\frac{1}{5}a^{2}-\frac{14}{15}a+\frac{11}{15}$, $\frac{11}{180}a^{7}-\frac{2}{45}a^{6}-\frac{2}{15}a^{5}+\frac{19}{90}a^{4}-\frac{7}{90}a^{3}+\frac{3}{5}a^{2}-\frac{4}{45}a+\frac{1}{45}$ | 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: | \( 319.6306375 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{3}\cdot 319.6306375 \cdot 1}{2\cdot\sqrt{19680620544}}\cr\approx \mathstrut & 1.130312701 \end{aligned}\]
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
\(\Q(\sqrt{3}) \) |
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 | R | ${\href{/padicField/5.6.0.1}{6} }{,}\,{\href{/padicField/5.2.0.1}{2} }$ | ${\href{/padicField/7.4.0.1}{4} }{,}\,{\href{/padicField/7.2.0.1}{2} }^{2}$ | ${\href{/padicField/11.3.0.1}{3} }^{2}{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ | R | ${\href{/padicField/17.4.0.1}{4} }{,}\,{\href{/padicField/17.2.0.1}{2} }^{2}$ | ${\href{/padicField/19.4.0.1}{4} }{,}\,{\href{/padicField/19.2.0.1}{2} }^{2}$ | ${\href{/padicField/23.4.0.1}{4} }{,}\,{\href{/padicField/23.3.0.1}{3} }{,}\,{\href{/padicField/23.1.0.1}{1} }$ | ${\href{/padicField/29.4.0.1}{4} }{,}\,{\href{/padicField/29.2.0.1}{2} }^{2}$ | ${\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.2.0.1}{2} }^{2}$ | ${\href{/padicField/37.4.0.1}{4} }{,}\,{\href{/padicField/37.3.0.1}{3} }{,}\,{\href{/padicField/37.1.0.1}{1} }$ | ${\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.2.0.1}{2} }$ | ${\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.2.0.1}{2} }$ | ${\href{/padicField/47.3.0.1}{3} }{,}\,{\href{/padicField/47.2.0.1}{2} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }$ | ${\href{/padicField/53.4.0.1}{4} }{,}\,{\href{/padicField/53.2.0.1}{2} }^{2}$ | ${\href{/padicField/59.4.0.1}{4} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.8.12.26 | $x^{8} + 2 x^{6} + 2 x^{5} + 2 x^{2} + 6$ | $8$ | $1$ | $12$ | $S_4\times C_2$ | $[4/3, 4/3, 2]_{3}^{2}$ |
\(3\) | 3.2.1.1 | $x^{2} + 6$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
3.6.6.7 | $x^{6} + 3 x + 6$ | $6$ | $1$ | $6$ | $C_3^2:D_4$ | $[5/4, 5/4]_{4}^{2}$ | |
\(13\) | $\Q_{13}$ | $x + 11$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{13}$ | $x + 11$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
13.2.0.1 | $x^{2} + 12 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
13.4.3.3 | $x^{4} + 26$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
1.39.2t1.a.a | $1$ | $ 3 \cdot 13 $ | \(\Q(\sqrt{-39}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
1.52.2t1.a.a | $1$ | $ 2^{2} \cdot 13 $ | \(\Q(\sqrt{-13}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
* | 1.12.2t1.a.a | $1$ | $ 2^{2} \cdot 3 $ | \(\Q(\sqrt{3}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
2.468.4t3.d.a | $2$ | $ 2^{2} \cdot 3^{2} \cdot 13 $ | 4.0.18252.2 | $D_{4}$ (as 4T3) | $1$ | $0$ | |
4.657072.6t13.c.a | $4$ | $ 2^{4} \cdot 3^{5} \cdot 13^{2}$ | 6.0.25625808.1 | $C_3^2:D_4$ (as 6T13) | $1$ | $0$ | |
4.34167744.12t34.h.a | $4$ | $ 2^{6} \cdot 3^{5} \cdot 13^{3}$ | 6.0.25625808.1 | $C_3^2:D_4$ (as 6T13) | $1$ | $-2$ | |
4.2628288.12t34.a.a | $4$ | $ 2^{6} \cdot 3^{5} \cdot 13^{2}$ | 6.0.25625808.1 | $C_3^2:D_4$ (as 6T13) | $1$ | $0$ | |
4.202176.6t13.a.a | $4$ | $ 2^{6} \cdot 3^{5} \cdot 13 $ | 6.0.25625808.1 | $C_3^2:D_4$ (as 6T13) | $1$ | $2$ | |
6.108...792.12t201.a.a | $6$ | $ 2^{10} \cdot 3^{7} \cdot 13^{6}$ | 8.2.19680620544.5 | $S_4\wr C_2$ (as 8T47) | $1$ | $2$ | |
6.277168739328.12t202.a.a | $6$ | $ 2^{10} \cdot 3^{6} \cdot 13^{5}$ | 8.2.19680620544.5 | $S_4\wr C_2$ (as 8T47) | $1$ | $0$ | |
* | 6.1640051712.8t47.e.a | $6$ | $ 2^{10} \cdot 3^{6} \cdot 13^{3}$ | 8.2.19680620544.5 | $S_4\wr C_2$ (as 8T47) | $1$ | $0$ |
6.378473472.12t200.a.a | $6$ | $ 2^{10} \cdot 3^{7} \cdot 13^{2}$ | 8.2.19680620544.5 | $S_4\wr C_2$ (as 8T47) | $1$ | $-2$ | |
9.256...592.16t1294.a.a | $9$ | $ 2^{12} \cdot 3^{10} \cdot 13^{9}$ | 8.2.19680620544.5 | $S_4\wr C_2$ (as 8T47) | $1$ | $-1$ | |
9.350...608.18t272.a.a | $9$ | $ 2^{12} \cdot 3^{11} \cdot 13^{6}$ | 8.2.19680620544.5 | $S_4\wr C_2$ (as 8T47) | $1$ | $1$ | |
9.186...576.18t273.c.a | $9$ | $ 2^{16} \cdot 3^{10} \cdot 13^{6}$ | 8.2.19680620544.5 | $S_4\wr C_2$ (as 8T47) | $1$ | $1$ | |
9.123...416.18t274.a.a | $9$ | $ 2^{16} \cdot 3^{11} \cdot 13^{9}$ | 8.2.19680620544.5 | $S_4\wr C_2$ (as 8T47) | $1$ | $-1$ | |
12.944...344.36t1763.b.a | $12$ | $ 2^{20} \cdot 3^{15} \cdot 13^{7}$ | 8.2.19680620544.5 | $S_4\wr C_2$ (as 8T47) | $1$ | $-2$ | |
12.159...136.24t2821.c.a | $12$ | $ 2^{20} \cdot 3^{15} \cdot 13^{9}$ | 8.2.19680620544.5 | $S_4\wr C_2$ (as 8T47) | $1$ | $2$ | |
18.431...928.36t1758.a.a | $18$ | $ 2^{28} \cdot 3^{22} \cdot 13^{15}$ | 8.2.19680620544.5 | $S_4\wr C_2$ (as 8T47) | $1$ | $0$ |