Normalized defining polynomial
\( x^{8} + 4x^{6} - 30x^{4} + 28x^{2} + 25 \)
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)
| |
Discriminant: | \(440301256704\) \(\medspace = 2^{26}\cdot 3^{8}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(28.54\) | 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: | $2^{55/16}3^{4/3}\approx 46.87618753574779$ | ||
Ramified primes: | \(2\), \(3\) | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2}a^{4}-\frac{1}{2}$, $\frac{1}{4}a^{5}-\frac{1}{4}a^{4}+\frac{1}{4}a-\frac{1}{4}$, $\frac{1}{12}a^{6}-\frac{1}{4}a^{4}-\frac{1}{4}a^{2}+\frac{1}{12}$, $\frac{1}{60}a^{7}-\frac{1}{10}a^{5}-\frac{1}{4}a^{4}-\frac{1}{4}a^{3}-\frac{1}{30}a-\frac{1}{4}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
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: | \( -1 \) (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: | $\frac{11}{60}a^{7}+\frac{13}{20}a^{5}-\frac{23}{4}a^{3}+\frac{443}{60}a+1$, $\frac{11}{60}a^{7}+\frac{13}{20}a^{5}-\frac{23}{4}a^{3}+\frac{443}{60}a-1$, $\frac{1}{12}a^{6}+\frac{3}{4}a^{4}-\frac{5}{4}a^{2}-\frac{11}{12}$ | 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: | \( 277.974030455 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{4}\cdot 277.974030455 \cdot 2}{2\cdot\sqrt{440301256704}}\cr\approx \mathstrut & 0.652903106618 \end{aligned}\]
Galois group
$C_2^3:S_4$ (as 8T39):
A solvable group of order 192 |
The 13 conjugacy class representatives for $C_2^3:S_4$ |
Character table for $C_2^3:S_4$ |
Intermediate fields
4.2.82944.2 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 8 siblings: | data not computed |
Degree 16 siblings: | data not computed |
Degree 24 siblings: | data not computed |
Degree 32 siblings: | data not computed |
Minimal sibling: | 8.4.27518828544.2 |
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 | ${\href{/padicField/5.6.0.1}{6} }{,}\,{\href{/padicField/5.2.0.1}{2} }$ | ${\href{/padicField/7.4.0.1}{4} }{,}\,{\href{/padicField/7.2.0.1}{2} }{,}\,{\href{/padicField/7.1.0.1}{1} }^{2}$ | ${\href{/padicField/11.4.0.1}{4} }^{2}$ | ${\href{/padicField/13.3.0.1}{3} }^{2}{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ | ${\href{/padicField/17.6.0.1}{6} }{,}\,{\href{/padicField/17.2.0.1}{2} }$ | ${\href{/padicField/19.4.0.1}{4} }{,}\,{\href{/padicField/19.2.0.1}{2} }{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | ${\href{/padicField/23.4.0.1}{4} }^{2}$ | ${\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.2.0.1}{2} }$ | ${\href{/padicField/31.4.0.1}{4} }^{2}$ | ${\href{/padicField/37.6.0.1}{6} }{,}\,{\href{/padicField/37.2.0.1}{2} }$ | ${\href{/padicField/41.4.0.1}{4} }^{2}$ | ${\href{/padicField/43.4.0.1}{4} }^{2}$ | ${\href{/padicField/47.4.0.1}{4} }^{2}$ | ${\href{/padicField/53.2.0.1}{2} }^{4}$ | ${\href{/padicField/59.2.0.1}{2} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }^{4}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.8.26.35 | $x^{8} + 4 x^{6} + 8 x^{3} + 10$ | $8$ | $1$ | $26$ | $(((C_4 \times C_2): C_2):C_2):C_2$ | $[2, 2, 3, 7/2, 4]^{2}$ |
\(3\) | 3.2.0.1 | $x^{2} + 2 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
3.6.8.5 | $x^{6} + 12 x^{5} + 57 x^{4} + 146 x^{3} + 240 x^{2} + 204 x + 65$ | $3$ | $2$ | $8$ | $S_3$ | $[2]^{2}$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
1.4.2t1.a.a | $1$ | $ 2^{2}$ | \(\Q(\sqrt{-1}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
2.324.3t2.b.a | $2$ | $ 2^{2} \cdot 3^{4}$ | 3.1.324.1 | $S_3$ (as 3T2) | $1$ | $0$ | |
3.5184.4t5.c.a | $3$ | $ 2^{6} \cdot 3^{4}$ | 4.2.5184.1 | $S_4$ (as 4T5) | $1$ | $1$ | |
3.82944.6t8.q.a | $3$ | $ 2^{10} \cdot 3^{4}$ | 4.2.82944.1 | $S_4$ (as 4T5) | $1$ | $-1$ | |
* | 3.82944.4t5.e.a | $3$ | $ 2^{10} \cdot 3^{4}$ | 4.2.82944.2 | $S_4$ (as 4T5) | $1$ | $1$ |
3.82944.6t8.p.a | $3$ | $ 2^{10} \cdot 3^{4}$ | 4.2.82944.2 | $S_4$ (as 4T5) | $1$ | $-1$ | |
3.82944.4t5.f.a | $3$ | $ 2^{10} \cdot 3^{4}$ | 4.2.82944.1 | $S_4$ (as 4T5) | $1$ | $1$ | |
3.20736.6t8.g.a | $3$ | $ 2^{8} \cdot 3^{4}$ | 4.2.5184.1 | $S_4$ (as 4T5) | $1$ | $-1$ | |
* | 4.5308416.8t39.q.a | $4$ | $ 2^{16} \cdot 3^{4}$ | 8.0.440301256704.5 | $C_2^3:S_4$ (as 8T39) | $1$ | $-2$ |
4.5308416.8t39.r.a | $4$ | $ 2^{16} \cdot 3^{4}$ | 8.0.440301256704.5 | $C_2^3:S_4$ (as 8T39) | $1$ | $2$ | |
6.1719926784.8t34.a.a | $6$ | $ 2^{18} \cdot 3^{8}$ | 8.0.6879707136.2 | $V_4^2:S_3$ (as 8T34) | $1$ | $0$ | |
8.228...536.24t333.i.a | $8$ | $ 2^{32} \cdot 3^{12}$ | 8.0.440301256704.5 | $C_2^3:S_4$ (as 8T39) | $1$ | $0$ |