Normalized defining polynomial
\( x^{8} - 7x^{5} - 7x^{4} + 42x^{3} - 14x^{2} - 28x + 28 \)
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: | \(-21346234560\) \(\medspace = -\,2^{6}\cdot 3^{4}\cdot 5\cdot 7^{7}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(19.55\) | 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: | $2^{3/2}3^{1/2}5^{1/2}7^{7/8}\approx 60.12449034589345$ | ||
Ramified primes: | \(2\), \(3\), \(5\), \(7\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{-35}) \) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}$, $\frac{1}{17102}a^{7}-\frac{3209}{17102}a^{6}-\frac{3137}{8551}a^{5}+\frac{4205}{17102}a^{4}-\frac{11}{503}a^{3}-\frac{5483}{17102}a^{2}+\frac{2763}{8551}a+\frac{906}{8551}$
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: | $\frac{2001}{17102}a^{7}+\frac{296}{8551}a^{6}-\frac{703}{8551}a^{5}-\frac{17081}{17102}a^{4}-\frac{1267}{1006}a^{3}+\frac{42480}{8551}a^{2}+\frac{13368}{8551}a-\frac{42661}{8551}$, $\frac{1055}{17102}a^{7}+\frac{701}{17102}a^{6}-\frac{298}{8551}a^{5}-\frac{10245}{17102}a^{4}-\frac{539}{503}a^{3}+\frac{30115}{17102}a^{2}+\frac{7625}{8551}a-\frac{10433}{8551}$, $\frac{341}{17102}a^{7}+\frac{259}{17102}a^{6}-\frac{842}{8551}a^{5}-\frac{2663}{17102}a^{4}-\frac{230}{503}a^{3}+\frac{28619}{17102}a^{2}+\frac{1573}{8551}a-\frac{15992}{8551}$, $\frac{10475}{17102}a^{7}+\frac{8257}{17102}a^{6}+\frac{1418}{8551}a^{5}-\frac{75785}{17102}a^{4}-\frac{4062}{503}a^{3}+\frac{353233}{17102}a^{2}+\frac{99902}{8551}a-\frac{112423}{8551}$ | 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: | \( 215.710864948 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{3}\cdot 215.710864948 \cdot 1}{2\cdot\sqrt{21346234560}}\cr\approx \mathstrut & 0.732455057906 \end{aligned}\]
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
\(\Q(\sqrt{21}) \) |
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 | R | R | R | ${\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.2.0.1}{2} }$ | ${\href{/padicField/13.6.0.1}{6} }{,}\,{\href{/padicField/13.2.0.1}{2} }$ | ${\href{/padicField/17.3.0.1}{3} }^{2}{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ | ${\href{/padicField/19.8.0.1}{8} }$ | ${\href{/padicField/23.8.0.1}{8} }$ | ${\href{/padicField/29.4.0.1}{4} }^{2}$ | ${\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.2.0.1}{2} }^{2}$ | ${\href{/padicField/37.4.0.1}{4} }{,}\,{\href{/padicField/37.3.0.1}{3} }{,}\,{\href{/padicField/37.1.0.1}{1} }$ | ${\href{/padicField/41.4.0.1}{4} }{,}\,{\href{/padicField/41.3.0.1}{3} }{,}\,{\href{/padicField/41.1.0.1}{1} }$ | ${\href{/padicField/43.3.0.1}{3} }{,}\,{\href{/padicField/43.2.0.1}{2} }{,}\,{\href{/padicField/43.1.0.1}{1} }^{3}$ | ${\href{/padicField/47.3.0.1}{3} }{,}\,{\href{/padicField/47.2.0.1}{2} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }$ | ${\href{/padicField/53.4.0.1}{4} }{,}\,{\href{/padicField/53.2.0.1}{2} }^{2}$ | ${\href{/padicField/59.2.0.1}{2} }^{3}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.4.6.1 | $x^{4} + 2 x^{3} + 31 x^{2} + 30 x + 183$ | $2$ | $2$ | $6$ | $C_2^2$ | $[3]^{2}$ |
2.4.0.1 | $x^{4} + x + 1$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
\(3\) | 3.2.1.1 | $x^{2} + 6$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
3.6.3.1 | $x^{6} + 18 x^{2} - 27$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
\(5\) | $\Q_{5}$ | $x + 3$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
5.2.0.1 | $x^{2} + 4 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
5.2.1.1 | $x^{2} + 5$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
5.3.0.1 | $x^{3} + 3 x + 3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
\(7\) | 7.8.7.1 | $x^{8} + 7$ | $8$ | $1$ | $7$ | $D_{8}$ | $[\ ]_{8}^{2}$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
1.35.2t1.a.a | $1$ | $ 5 \cdot 7 $ | \(\Q(\sqrt{-35}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
1.15.2t1.a.a | $1$ | $ 3 \cdot 5 $ | \(\Q(\sqrt{-15}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
* | 1.21.2t1.a.a | $1$ | $ 3 \cdot 7 $ | \(\Q(\sqrt{21}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
2.47040.4t3.b.a | $2$ | $ 2^{6} \cdot 3 \cdot 5 \cdot 7^{2}$ | 4.0.1646400.5 | $D_{4}$ (as 4T3) | $1$ | $0$ | |
4.1646400.6t13.b.a | $4$ | $ 2^{6} \cdot 3 \cdot 5^{2} \cdot 7^{3}$ | 6.0.57624000.1 | $C_3^2:D_4$ (as 6T13) | $1$ | $0$ | |
4.24696000.12t34.b.a | $4$ | $ 2^{6} \cdot 3^{2} \cdot 5^{3} \cdot 7^{3}$ | 6.0.57624000.1 | $C_3^2:D_4$ (as 6T13) | $1$ | $-2$ | |
4.14817600.12t34.h.a | $4$ | $ 2^{6} \cdot 3^{3} \cdot 5^{2} \cdot 7^{3}$ | 6.0.57624000.1 | $C_3^2:D_4$ (as 6T13) | $1$ | $0$ | |
4.987840.6t13.b.a | $4$ | $ 2^{6} \cdot 3^{2} \cdot 5 \cdot 7^{3}$ | 6.0.57624000.1 | $C_3^2:D_4$ (as 6T13) | $1$ | $2$ | |
6.116...000.12t201.a.a | $6$ | $ 2^{12} \cdot 3^{3} \cdot 5^{4} \cdot 7^{5}$ | 8.2.21346234560.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $2$ | |
6.635304600000.12t202.a.a | $6$ | $ 2^{6} \cdot 3^{3} \cdot 5^{5} \cdot 7^{6}$ | 8.2.21346234560.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $0$ | |
* | 6.1016487360.8t47.a.a | $6$ | $ 2^{6} \cdot 3^{3} \cdot 5 \cdot 7^{6}$ | 8.2.21346234560.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $0$ |
6.46467993600.12t200.a.a | $6$ | $ 2^{12} \cdot 3^{3} \cdot 5^{2} \cdot 7^{5}$ | 8.2.21346234560.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $-2$ | |
9.796...000.16t1294.a.a | $9$ | $ 2^{12} \cdot 3^{3} \cdot 5^{3} \cdot 7^{8}$ | 8.2.21346234560.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $-1$ | |
9.142...000.18t272.a.a | $9$ | $ 2^{12} \cdot 3^{3} \cdot 5^{6} \cdot 7^{7}$ | 8.2.21346234560.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $1$ | |
9.268...000.18t273.a.a | $9$ | $ 2^{12} \cdot 3^{6} \cdot 5^{6} \cdot 7^{8}$ | 8.2.21346234560.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $1$ | |
9.307...000.18t274.a.a | $9$ | $ 2^{12} \cdot 3^{6} \cdot 5^{3} \cdot 7^{7}$ | 8.2.21346234560.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $-1$ | |
12.295...000.36t1763.a.a | $12$ | $ 2^{18} \cdot 3^{6} \cdot 5^{7} \cdot 7^{11}$ | 8.2.21346234560.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $-2$ | |
12.118...000.24t2821.a.a | $12$ | $ 2^{18} \cdot 3^{6} \cdot 5^{5} \cdot 7^{11}$ | 8.2.21346234560.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $2$ | |
18.137...000.36t1758.a.a | $18$ | $ 2^{30} \cdot 3^{9} \cdot 5^{9} \cdot 7^{16}$ | 8.2.21346234560.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $0$ |