Normalized defining polynomial
\( x^{8} - 4x^{7} + 8x^{5} + 13x^{4} - 18x^{3} - 18x^{2} + 7 \)
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: | \(-14139741952\) \(\medspace = -\,2^{8}\cdot 7\cdot 53^{4}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(18.57\) | 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: | $2^{31/24}7^{1/2}53^{1/2}\approx 47.153872609627676$ | ||
Ramified primes: | \(2\), \(7\), \(53\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{-7}) \) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{37}a^{7}+\frac{14}{37}a^{6}-\frac{7}{37}a^{5}-\frac{7}{37}a^{4}-\frac{2}{37}a^{3}-\frac{17}{37}a^{2}+\frac{9}{37}a+\frac{14}{37}$
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: | $a-2$, $\frac{18}{37}a^{7}-\frac{44}{37}a^{6}-\frac{52}{37}a^{5}+\frac{22}{37}a^{4}+\frac{223}{37}a^{3}+\frac{64}{37}a^{2}-\frac{97}{37}a-\frac{81}{37}$, $\frac{4}{37}a^{7}-\frac{18}{37}a^{6}+\frac{9}{37}a^{5}+\frac{9}{37}a^{4}+\frac{66}{37}a^{3}-\frac{31}{37}a^{2}-\frac{75}{37}a-\frac{55}{37}$, $\frac{9}{37}a^{7}-\frac{22}{37}a^{6}-\frac{26}{37}a^{5}+\frac{11}{37}a^{4}+\frac{93}{37}a^{3}+\frac{69}{37}a^{2}-\frac{30}{37}a-\frac{22}{37}$ | 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: | \( 132.403904441 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{3}\cdot 132.403904441 \cdot 1}{2\cdot\sqrt{14139741952}}\cr\approx \mathstrut & 0.552394888206 \end{aligned}\]
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
\(\Q(\sqrt{53}) \) |
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 | ${\href{/padicField/3.4.0.1}{4} }{,}\,{\href{/padicField/3.2.0.1}{2} }^{2}$ | ${\href{/padicField/5.8.0.1}{8} }$ | R | ${\href{/padicField/11.4.0.1}{4} }{,}\,{\href{/padicField/11.2.0.1}{2} }{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ | ${\href{/padicField/13.4.0.1}{4} }{,}\,{\href{/padicField/13.3.0.1}{3} }{,}\,{\href{/padicField/13.1.0.1}{1} }$ | ${\href{/padicField/17.3.0.1}{3} }{,}\,{\href{/padicField/17.2.0.1}{2} }{,}\,{\href{/padicField/17.1.0.1}{1} }^{3}$ | ${\href{/padicField/19.8.0.1}{8} }$ | ${\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.2.0.1}{2} }$ | ${\href{/padicField/29.3.0.1}{3} }{,}\,{\href{/padicField/29.2.0.1}{2} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }$ | ${\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.2.0.1}{2} }^{2}$ | ${\href{/padicField/37.4.0.1}{4} }{,}\,{\href{/padicField/37.2.0.1}{2} }{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.8.0.1}{8} }$ | ${\href{/padicField/43.4.0.1}{4} }{,}\,{\href{/padicField/43.2.0.1}{2} }{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | ${\href{/padicField/47.3.0.1}{3} }{,}\,{\href{/padicField/47.2.0.1}{2} }{,}\,{\href{/padicField/47.1.0.1}{1} }^{3}$ | R | ${\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.3.0.1}{3} }{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.8.8.12 | $x^{8} - 2 x^{6} - 2 x^{5} + 8 x^{4} + 8 x^{3} - 4 x + 4$ | $4$ | $2$ | $8$ | $A_4\wr C_2$ | $[4/3, 4/3, 4/3, 4/3]_{3}^{6}$ |
\(7\) | $\Q_{7}$ | $x + 4$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
7.2.1.1 | $x^{2} + 21$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
7.2.0.1 | $x^{2} + 6 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
7.3.0.1 | $x^{3} + 6 x^{2} + 4$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
\(53\) | 53.2.1.1 | $x^{2} + 53$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
53.6.3.1 | $x^{6} + 7314 x^{5} + 17831697 x^{4} + 14491896314 x^{3} + 998947242 x^{2} + 44403234204 x + 739971484841$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
1.7.2t1.a.a | $1$ | $ 7 $ | \(\Q(\sqrt{-7}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
1.371.2t1.a.a | $1$ | $ 7 \cdot 53 $ | \(\Q(\sqrt{-371}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
* | 1.53.2t1.a.a | $1$ | $ 53 $ | \(\Q(\sqrt{53}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
2.371.4t3.a.a | $2$ | $ 7 \cdot 53 $ | 4.0.2597.2 | $D_{4}$ (as 4T3) | $1$ | $0$ | |
4.29179892.12t34.b.a | $4$ | $ 2^{2} \cdot 7^{2} \cdot 53^{3}$ | 6.0.72716.1 | $C_3^2:D_4$ (as 6T13) | $1$ | $0$ | |
4.15415792.12t34.b.a | $4$ | $ 2^{4} \cdot 7^{3} \cdot 53^{2}$ | 6.0.72716.1 | $C_3^2:D_4$ (as 6T13) | $1$ | $-2$ | |
4.10388.6t13.b.a | $4$ | $ 2^{2} \cdot 7^{2} \cdot 53 $ | 6.0.72716.1 | $C_3^2:D_4$ (as 6T13) | $1$ | $0$ | |
4.314608.6t13.b.a | $4$ | $ 2^{4} \cdot 7 \cdot 53^{2}$ | 6.0.72716.1 | $C_3^2:D_4$ (as 6T13) | $1$ | $2$ | |
6.91508141312.12t201.a.a | $6$ | $ 2^{8} \cdot 7^{4} \cdot 53^{3}$ | 8.2.14139741952.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $2$ | |
6.640556989184.12t202.a.a | $6$ | $ 2^{8} \cdot 7^{5} \cdot 53^{3}$ | 8.2.14139741952.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $0$ | |
* | 6.266787584.8t47.a.a | $6$ | $ 2^{8} \cdot 7 \cdot 53^{3}$ | 8.2.14139741952.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $0$ |
6.1867513088.12t200.a.a | $6$ | $ 2^{8} \cdot 7^{2} \cdot 53^{3}$ | 8.2.14139741952.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $-2$ | |
9.209161465856.16t1294.a.a | $9$ | $ 2^{12} \cdot 7^{3} \cdot 53^{3}$ | 8.2.14139741952.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $-1$ | |
9.717...608.18t272.a.a | $9$ | $ 2^{12} \cdot 7^{6} \cdot 53^{3}$ | 8.2.14139741952.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $1$ | |
9.106...216.18t273.a.a | $9$ | $ 2^{12} \cdot 7^{6} \cdot 53^{6}$ | 8.2.14139741952.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $1$ | |
9.311...712.18t274.a.a | $9$ | $ 2^{12} \cdot 7^{3} \cdot 53^{6}$ | 8.2.14139741952.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $-1$ | |
12.119...192.36t1763.a.a | $12$ | $ 2^{16} \cdot 7^{7} \cdot 53^{6}$ | 8.2.14139741952.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $-2$ | |
12.244...208.24t2821.a.a | $12$ | $ 2^{16} \cdot 7^{5} \cdot 53^{6}$ | 8.2.14139741952.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $2$ | |
18.223...896.36t1758.a.a | $18$ | $ 2^{24} \cdot 7^{9} \cdot 53^{9}$ | 8.2.14139741952.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $0$ |