Normalized defining polynomial
\( x^{8} + 8x^{6} - 40x^{2} + 100 \)
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: | \(539343360000\) \(\medspace = 2^{12}\cdot 3^{6}\cdot 5^{4}\cdot 17^{2}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(29.27\) | 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: | $2^{3/2}3^{3/4}5^{3/4}17^{1/2}\approx 88.88692832055303$ | ||
Ramified primes: | \(2\), \(3\), \(5\), \(17\) | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{4}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{8}a^{5}+\frac{1}{4}a^{3}-\frac{1}{4}a-\frac{1}{2}$, $\frac{1}{40}a^{6}-\frac{1}{20}a^{4}-\frac{1}{4}a^{2}-\frac{1}{2}a$, $\frac{1}{40}a^{7}-\frac{1}{20}a^{5}-\frac{1}{4}a^{3}-\frac{1}{2}a^{2}$
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: | $\frac{1}{10}a^{7}-\frac{1}{8}a^{6}+\frac{37}{40}a^{5}-\frac{5}{4}a^{4}+\frac{5}{4}a^{3}-\frac{7}{4}a^{2}-\frac{11}{4}a+\frac{13}{2}$, $\frac{1}{10}a^{7}+\frac{1}{8}a^{6}+\frac{37}{40}a^{5}+\frac{5}{4}a^{4}+\frac{5}{4}a^{3}+\frac{7}{4}a^{2}-\frac{11}{4}a-\frac{13}{2}$, $\frac{1}{2}a^{4}+4a^{2}+8$ | 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: | \( 509.870154164 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{4}\cdot 509.870154164 \cdot 2}{2\cdot\sqrt{539343360000}}\cr\approx \mathstrut & 1.08204767456 \end{aligned}\]
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
4.2.146880.1 |
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: | 8.0.210681000000.2 |
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 | R | ${\href{/padicField/7.6.0.1}{6} }{,}\,{\href{/padicField/7.2.0.1}{2} }$ | ${\href{/padicField/11.3.0.1}{3} }^{2}{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ | ${\href{/padicField/13.4.0.1}{4} }^{2}$ | R | ${\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.2.0.1}{2} }$ | ${\href{/padicField/23.4.0.1}{4} }{,}\,{\href{/padicField/23.2.0.1}{2} }{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ | ${\href{/padicField/29.3.0.1}{3} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.4.0.1}{4} }^{2}$ | ${\href{/padicField/37.3.0.1}{3} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.3.0.1}{3} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.4.0.1}{4} }^{2}$ | ${\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.2.0.1}{2} }$ | ${\href{/padicField/53.3.0.1}{3} }^{2}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | ${\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.2.0.1}{2} }{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.8.12.16 | $x^{8} + 8 x^{7} + 32 x^{6} + 78 x^{5} + 137 x^{4} + 186 x^{3} + 128 x^{2} - 10 x + 7$ | $4$ | $2$ | $12$ | $A_4\times C_2$ | $[2, 2]^{6}$ |
\(3\) | 3.4.3.2 | $x^{4} + 6$ | $4$ | $1$ | $3$ | $D_{4}$ | $[\ ]_{4}^{2}$ |
3.4.3.2 | $x^{4} + 6$ | $4$ | $1$ | $3$ | $D_{4}$ | $[\ ]_{4}^{2}$ | |
\(5\) | 5.2.1.2 | $x^{2} + 10$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
5.2.0.1 | $x^{2} + 4 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
5.4.3.1 | $x^{4} + 20$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ | |
\(17\) | 17.2.0.1 | $x^{2} + 16 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
17.2.0.1 | $x^{2} + 16 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
17.4.2.1 | $x^{4} + 338 x^{3} + 31049 x^{2} + 420472 x + 123735$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
1.255.2t1.a.a | $1$ | $ 3 \cdot 5 \cdot 17 $ | \(\Q(\sqrt{-255}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
2.255.3t2.a.a | $2$ | $ 3 \cdot 5 \cdot 17 $ | 3.1.255.1 | $S_3$ (as 3T2) | $1$ | $0$ | |
3.57375.4t5.a.a | $3$ | $ 3^{3} \cdot 5^{3} \cdot 17 $ | 4.2.57375.1 | $S_4$ (as 4T5) | $1$ | $1$ | |
3.4161600.6t8.c.a | $3$ | $ 2^{6} \cdot 3^{2} \cdot 5^{2} \cdot 17^{2}$ | 4.2.408000.2 | $S_4$ (as 4T5) | $1$ | $-1$ | |
* | 3.146880.4t5.a.a | $3$ | $ 2^{6} \cdot 3^{3} \cdot 5 \cdot 17 $ | 4.2.146880.1 | $S_4$ (as 4T5) | $1$ | $1$ |
3.4161600.6t8.a.a | $3$ | $ 2^{6} \cdot 3^{2} \cdot 5^{2} \cdot 17^{2}$ | 4.2.146880.1 | $S_4$ (as 4T5) | $1$ | $-1$ | |
3.408000.4t5.a.a | $3$ | $ 2^{6} \cdot 3 \cdot 5^{3} \cdot 17 $ | 4.2.408000.2 | $S_4$ (as 4T5) | $1$ | $1$ | |
3.65025.6t8.a.a | $3$ | $ 3^{2} \cdot 5^{2} \cdot 17^{2}$ | 4.2.57375.1 | $S_4$ (as 4T5) | $1$ | $-1$ | |
* | 4.3672000.8t39.w.a | $4$ | $ 2^{6} \cdot 3^{3} \cdot 5^{3} \cdot 17 $ | 8.0.539343360000.1 | $C_2^3:S_4$ (as 8T39) | $1$ | $-2$ |
4.1061208000.8t39.w.a | $4$ | $ 2^{6} \cdot 3^{3} \cdot 5^{3} \cdot 17^{3}$ | 8.0.539343360000.1 | $C_2^3:S_4$ (as 8T39) | $1$ | $2$ | |
6.152...000.8t34.a.a | $6$ | $ 2^{12} \cdot 3^{5} \cdot 5^{5} \cdot 17^{3}$ | 8.0.3896755776000000.32 | $V_4^2:S_3$ (as 8T34) | $1$ | $0$ | |
8.389...000.24t333.w.a | $8$ | $ 2^{12} \cdot 3^{6} \cdot 5^{6} \cdot 17^{4}$ | 8.0.539343360000.1 | $C_2^3:S_4$ (as 8T39) | $1$ | $0$ |