Normalized defining polynomial
\( x^{8} - 4x^{7} + 8x^{6} - 7x^{4} - 12x^{3} + 50x^{2} + 80x + 28 \)
Invariants
Degree: | $8$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[0, 4]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(9475854336\) \(\medspace = 2^{12}\cdot 3^{4}\cdot 13^{4}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(17.66\) | 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))
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Galois root discriminant: | $2^{3/2}3^{1/2}13^{3/4}\approx 33.54000593239012$ | ||
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: | \(\Q\) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $\frac{1}{2}a^{3}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{4}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{5}-\frac{1}{2}a^{2}$, $\frac{1}{12}a^{6}-\frac{1}{6}a^{5}-\frac{1}{4}a^{4}+\frac{1}{6}a^{3}-\frac{1}{2}a^{2}+\frac{1}{3}a+\frac{1}{3}$, $\frac{1}{24}a^{7}-\frac{1}{24}a^{6}+\frac{1}{24}a^{5}-\frac{1}{24}a^{4}+\frac{1}{12}a^{3}-\frac{1}{12}a^{2}-\frac{1}{6}a+\frac{1}{6}$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $2$ |
Class group and class number
$C_{2}$, which has order $2$
Unit group
Rank: | $3$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
Torsion generator: | \( \frac{31}{12} a^{7} - 12 a^{6} + \frac{341}{12} a^{5} - \frac{55}{3} a^{4} - \frac{37}{6} a^{3} - \frac{163}{6} a^{2} + 147 a + \frac{338}{3} \) (order $6$) | 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: | $\frac{53}{24}a^{7}-\frac{247}{24}a^{6}+\frac{195}{8}a^{5}-\frac{383}{24}a^{4}-\frac{21}{4}a^{3}-\frac{281}{12}a^{2}+\frac{749}{6}a+\frac{191}{2}$, $\frac{5}{24}a^{7}-\frac{23}{24}a^{6}+\frac{53}{24}a^{5}-\frac{35}{24}a^{4}-\frac{7}{12}a^{3}-\frac{29}{12}a^{2}+\frac{67}{6}a+\frac{53}{6}$, $\frac{1}{6}a^{7}-\frac{2}{3}a^{6}+\frac{7}{6}a^{5}+\frac{5}{6}a^{4}-\frac{19}{6}a^{3}-\frac{4}{3}a^{2}+\frac{37}{3}a+\frac{23}{3}$ | sage: UK.fundamental_units()
gp: K.fu
magma: [K|fUK(g): g in Generators(UK)];
oscar: [K(fUK(a)) for a in gens(UK)]
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Regulator: | \( 126.444530989 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{4}\cdot 126.444530989 \cdot 2}{6\cdot\sqrt{9475854336}}\cr\approx \mathstrut & 0.674821763635 \end{aligned}\]
Galois group
$C_2^3:C_4$ (as 8T19):
A solvable group of order 32 |
The 11 conjugacy class representatives for $C_2^3 : C_4 $ |
Character table for $C_2^3 : C_4 $ |
Intermediate fields
\(\Q(\sqrt{-3}) \), 4.0.7488.2 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Galois closure: | data not computed |
Degree 8 siblings: | data not computed |
Degree 16 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 | ${\href{/padicField/5.4.0.1}{4} }^{2}$ | ${\href{/padicField/7.4.0.1}{4} }{,}\,{\href{/padicField/7.2.0.1}{2} }{,}\,{\href{/padicField/7.1.0.1}{1} }^{2}$ | ${\href{/padicField/11.4.0.1}{4} }^{2}$ | R | ${\href{/padicField/17.4.0.1}{4} }^{2}$ | ${\href{/padicField/19.4.0.1}{4} }{,}\,{\href{/padicField/19.2.0.1}{2} }{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | ${\href{/padicField/23.2.0.1}{2} }^{4}$ | ${\href{/padicField/29.4.0.1}{4} }^{2}$ | ${\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.2.0.1}{2} }{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.4.0.1}{4} }{,}\,{\href{/padicField/37.2.0.1}{2} }{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.4.0.1}{4} }^{2}$ | ${\href{/padicField/43.2.0.1}{2} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }^{4}$ | ${\href{/padicField/47.4.0.1}{4} }^{2}$ | ${\href{/padicField/53.4.0.1}{4} }^{2}$ | ${\href{/padicField/59.4.0.1}{4} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.4.6.4 | $x^{4} + 4 x^{3} + 24 x^{2} + 88 x + 124$ | $2$ | $2$ | $6$ | $C_4$ | $[3]^{2}$ |
2.4.6.4 | $x^{4} + 4 x^{3} + 24 x^{2} + 88 x + 124$ | $2$ | $2$ | $6$ | $C_4$ | $[3]^{2}$ | |
\(3\) | 3.4.2.1 | $x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
3.4.2.1 | $x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
\(13\) | 13.2.0.1 | $x^{2} + 12 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
13.2.1.2 | $x^{2} + 26$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
13.4.3.2 | $x^{4} + 13$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
* | 1.3.2t1.a.a | $1$ | $ 3 $ | \(\Q(\sqrt{-3}) \) | $C_2$ (as 2T1) | $1$ | $-1$ |
1.39.2t1.a.a | $1$ | $ 3 \cdot 13 $ | \(\Q(\sqrt{-39}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
1.13.2t1.a.a | $1$ | $ 13 $ | \(\Q(\sqrt{13}) \) | $C_2$ (as 2T1) | $1$ | $1$ | |
1.312.4t1.b.a | $1$ | $ 2^{3} \cdot 3 \cdot 13 $ | 4.0.1265472.2 | $C_4$ (as 4T1) | $0$ | $-1$ | |
1.104.4t1.a.a | $1$ | $ 2^{3} \cdot 13 $ | 4.4.140608.1 | $C_4$ (as 4T1) | $0$ | $1$ | |
1.104.4t1.a.b | $1$ | $ 2^{3} \cdot 13 $ | 4.4.140608.1 | $C_4$ (as 4T1) | $0$ | $1$ | |
1.312.4t1.b.b | $1$ | $ 2^{3} \cdot 3 \cdot 13 $ | 4.0.1265472.2 | $C_4$ (as 4T1) | $0$ | $-1$ | |
2.507.4t3.c.a | $2$ | $ 3 \cdot 13^{2}$ | 4.0.19773.1 | $D_{4}$ (as 4T3) | $1$ | $0$ | |
* | 2.2496.4t3.j.a | $2$ | $ 2^{6} \cdot 3 \cdot 13 $ | 4.0.7488.2 | $D_{4}$ (as 4T3) | $1$ | $0$ |
* | 4.1265472.8t19.d.a | $4$ | $ 2^{6} \cdot 3^{2} \cdot 13^{3}$ | 8.0.9475854336.14 | $C_2^3 : C_4 $ (as 8T19) | $1$ | $0$ |