Normalized defining polynomial
\( x^{8} - 4x^{7} + 10x^{6} + 3x^{5} - 19x^{4} + 41x^{3} - 26x^{2} + 15x + 1 \)
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: | \(-248438446096\) \(\medspace = -\,2^{4}\cdot 353^{4}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(26.57\) | 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: | $2^{3/2}353^{1/2}\approx 53.14132102234569$ | ||
Ramified primes: | \(2\), \(353\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{-1}) \) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{521}a^{7}+\frac{147}{521}a^{6}-\frac{196}{521}a^{5}+\frac{104}{521}a^{4}+\frac{55}{521}a^{3}+\frac{10}{521}a^{2}-\frac{79}{521}a+\frac{69}{521}$
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: | $a$, $\frac{74}{521}a^{7}-\frac{584}{521}a^{6}+\frac{1647}{521}a^{5}-\frac{1682}{521}a^{4}-\frac{4787}{521}a^{3}+\frac{7513}{521}a^{2}-\frac{7409}{521}a-\frac{625}{521}$, $\frac{164}{521}a^{7}-\frac{379}{521}a^{6}+\frac{679}{521}a^{5}+\frac{2468}{521}a^{4}-\frac{358}{521}a^{3}+\frac{1640}{521}a^{2}+\frac{590}{521}a+\frac{375}{521}$, $\frac{614}{521}a^{7}-\frac{2480}{521}a^{6}+\frac{6259}{521}a^{5}+\frac{1336}{521}a^{4}-\frac{11557}{521}a^{3}+\frac{25417}{521}a^{2}-\frac{17246}{521}a+\frac{10064}{521}$ | 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: | \( 1058.88591004 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{3}\cdot 1058.88591004 \cdot 1}{2\cdot\sqrt{248438446096}}\cr\approx \mathstrut & 1.05392417882 \end{aligned}\]
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
\(\Q(\sqrt{353}) \) |
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 | ${\href{/padicField/3.4.0.1}{4} }{,}\,{\href{/padicField/3.2.0.1}{2} }^{2}$ | ${\href{/padicField/5.6.0.1}{6} }{,}\,{\href{/padicField/5.2.0.1}{2} }$ | ${\href{/padicField/7.8.0.1}{8} }$ | ${\href{/padicField/11.3.0.1}{3} }{,}\,{\href{/padicField/11.2.0.1}{2} }{,}\,{\href{/padicField/11.1.0.1}{1} }^{3}$ | ${\href{/padicField/13.6.0.1}{6} }{,}\,{\href{/padicField/13.2.0.1}{2} }$ | ${\href{/padicField/17.3.0.1}{3} }{,}\,{\href{/padicField/17.2.0.1}{2} }^{2}{,}\,{\href{/padicField/17.1.0.1}{1} }$ | ${\href{/padicField/19.3.0.1}{3} }{,}\,{\href{/padicField/19.2.0.1}{2} }{,}\,{\href{/padicField/19.1.0.1}{1} }^{3}$ | ${\href{/padicField/23.4.0.1}{4} }{,}\,{\href{/padicField/23.3.0.1}{3} }{,}\,{\href{/padicField/23.1.0.1}{1} }$ | ${\href{/padicField/29.4.0.1}{4} }{,}\,{\href{/padicField/29.2.0.1}{2} }{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.2.0.1}{2} }^{2}$ | ${\href{/padicField/37.6.0.1}{6} }{,}\,{\href{/padicField/37.2.0.1}{2} }$ | ${\href{/padicField/41.4.0.1}{4} }{,}\,{\href{/padicField/41.2.0.1}{2} }{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.4.0.1}{4} }{,}\,{\href{/padicField/43.3.0.1}{3} }{,}\,{\href{/padicField/43.1.0.1}{1} }$ | ${\href{/padicField/47.3.0.1}{3} }{,}\,{\href{/padicField/47.2.0.1}{2} }{,}\,{\href{/padicField/47.1.0.1}{1} }^{3}$ | ${\href{/padicField/53.4.0.1}{4} }^{2}$ | ${\href{/padicField/59.8.0.1}{8} }$ |
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 | $[\ ]$ |
2.3.0.1 | $x^{3} + x + 1$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
2.4.4.3 | $x^{4} + 2 x^{3} + 4 x^{2} + 12 x + 12$ | $2$ | $2$ | $4$ | $D_{4}$ | $[2, 2]^{2}$ | |
\(353\) | Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
Deg $6$ | $2$ | $3$ | $3$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
1.4.2t1.a.a | $1$ | $ 2^{2}$ | \(\Q(\sqrt{-1}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
1.1412.2t1.a.a | $1$ | $ 2^{2} \cdot 353 $ | \(\Q(\sqrt{-353}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
* | 1.353.2t1.a.a | $1$ | $ 353 $ | \(\Q(\sqrt{353}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
2.1412.4t3.c.a | $2$ | $ 2^{2} \cdot 353 $ | 4.0.5648.1 | $D_{4}$ (as 4T3) | $1$ | $0$ | |
4.703791632.12t34.b.a | $4$ | $ 2^{4} \cdot 353^{3}$ | 6.0.22592.1 | $C_3^2:D_4$ (as 6T13) | $1$ | $0$ | |
4.7974976.12t34.b.a | $4$ | $ 2^{6} \cdot 353^{2}$ | 6.0.22592.1 | $C_3^2:D_4$ (as 6T13) | $1$ | $-2$ | |
4.5648.6t13.b.a | $4$ | $ 2^{4} \cdot 353 $ | 6.0.22592.1 | $C_3^2:D_4$ (as 6T13) | $1$ | $0$ | |
4.498436.6t13.b.a | $4$ | $ 2^{2} \cdot 353^{2}$ | 6.0.22592.1 | $C_3^2:D_4$ (as 6T13) | $1$ | $2$ | |
6.45042664448.12t201.b.a | $6$ | $ 2^{10} \cdot 353^{3}$ | 8.2.248438446096.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $2$ | |
6.180170657792.12t202.b.a | $6$ | $ 2^{12} \cdot 353^{3}$ | 8.2.248438446096.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $0$ | |
* | 6.703791632.8t47.a.a | $6$ | $ 2^{4} \cdot 353^{3}$ | 8.2.248438446096.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $0$ |
6.2815166528.12t200.a.a | $6$ | $ 2^{6} \cdot 353^{3}$ | 8.2.248438446096.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $-2$ | |
9.180170657792.16t1294.a.a | $9$ | $ 2^{12} \cdot 353^{3}$ | 8.2.248438446096.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $-1$ | |
9.115...688.18t272.b.a | $9$ | $ 2^{18} \cdot 353^{3}$ | 8.2.248438446096.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $1$ | |
9.507...176.18t273.b.a | $9$ | $ 2^{18} \cdot 353^{6}$ | 8.2.248438446096.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $1$ | |
9.792...784.18t274.a.a | $9$ | $ 2^{12} \cdot 353^{6}$ | 8.2.248438446096.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $-1$ | |
12.507...176.36t1763.a.a | $12$ | $ 2^{18} \cdot 353^{6}$ | 8.2.248438446096.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $-2$ | |
12.317...136.24t2821.a.a | $12$ | $ 2^{14} \cdot 353^{6}$ | 8.2.248438446096.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $2$ | |
18.913...392.36t1758.a.a | $18$ | $ 2^{30} \cdot 353^{9}$ | 8.2.248438446096.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $0$ |