Normalized defining polynomial
\( x^{8} - 4x^{7} + 12x^{6} - 2x^{5} - 3x^{4} + 38x^{3} + 8x^{2} - 14x + 1 \)
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: | \(-170984391424\) \(\medspace = -\,2^{8}\cdot 7^{4}\cdot 11^{4}\cdot 19\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(25.36\) | 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: | $2\cdot 7^{1/2}11^{1/2}19^{1/2}\approx 76.49836599562111$ | ||
Ramified primes: | \(2\), \(7\), \(11\), \(19\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{-19}) \) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2}a^{4}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{2}a^{6}-\frac{1}{2}$, $\frac{1}{74}a^{7}+\frac{7}{37}a^{6}+\frac{5}{74}a^{5}+\frac{7}{37}a^{4}+\frac{27}{74}a^{3}+\frac{3}{37}a^{2}-\frac{16}{37}a+\frac{1}{37}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{2}$, which has order $2$
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: | $\frac{43}{74}a^{7}-\frac{175}{74}a^{6}+\frac{274}{37}a^{5}-\frac{106}{37}a^{4}+\frac{81}{37}a^{3}+\frac{721}{37}a^{2}+\frac{363}{74}a-\frac{25}{74}$, $a$, $a+1$, $\frac{8}{37}a^{7}-\frac{36}{37}a^{6}+\frac{114}{37}a^{5}-\frac{73}{37}a^{4}+\frac{31}{37}a^{3}+\frac{233}{37}a^{2}+\frac{77}{37}a+\frac{16}{37}$ | 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: | \( 158.082867494 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{3}\cdot 158.082867494 \cdot 2}{2\cdot\sqrt{170984391424}}\cr\approx \mathstrut & 0.379320372163 \end{aligned}\]
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
\(\Q(\sqrt{77}) \) |
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 | ${\href{/padicField/3.8.0.1}{8} }$ | ${\href{/padicField/5.6.0.1}{6} }{,}\,{\href{/padicField/5.2.0.1}{2} }$ | R | R | ${\href{/padicField/13.3.0.1}{3} }{,}\,{\href{/padicField/13.2.0.1}{2} }{,}\,{\href{/padicField/13.1.0.1}{1} }^{3}$ | ${\href{/padicField/17.4.0.1}{4} }{,}\,{\href{/padicField/17.2.0.1}{2} }{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ | R | ${\href{/padicField/23.3.0.1}{3} }{,}\,{\href{/padicField/23.2.0.1}{2} }^{2}{,}\,{\href{/padicField/23.1.0.1}{1} }$ | ${\href{/padicField/29.8.0.1}{8} }$ | ${\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.2.0.1}{2} }^{2}$ | ${\href{/padicField/37.3.0.1}{3} }{,}\,{\href{/padicField/37.2.0.1}{2} }{,}\,{\href{/padicField/37.1.0.1}{1} }^{3}$ | ${\href{/padicField/41.4.0.1}{4} }{,}\,{\href{/padicField/41.3.0.1}{3} }{,}\,{\href{/padicField/41.1.0.1}{1} }$ | ${\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.2.0.1}{2} }$ | ${\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.2.0.1}{2} }$ | ${\href{/padicField/53.3.0.1}{3} }{,}\,{\href{/padicField/53.2.0.1}{2} }{,}\,{\href{/padicField/53.1.0.1}{1} }^{3}$ | ${\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.2.0.1}{2} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.8.8.4 | $x^{8} + 8 x^{7} + 20 x^{6} + 8 x^{5} + 32 x^{4} + 224 x^{3} + 144 x^{2} - 224 x + 752$ | $2$ | $4$ | $8$ | $C_8$ | $[2]^{4}$ |
\(7\) | 7.2.1.1 | $x^{2} + 21$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
7.6.3.1 | $x^{6} + 861 x^{5} + 33033 x^{4} + 1385475 x^{3} + 277830 x^{2} + 8232 x - 1372$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
\(11\) | 11.2.1.2 | $x^{2} + 11$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
11.6.3.2 | $x^{6} + 37 x^{4} + 18 x^{3} + 367 x^{2} - 558 x + 972$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
\(19\) | $\Q_{19}$ | $x + 17$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
19.2.0.1 | $x^{2} + 18 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
19.2.1.1 | $x^{2} + 38$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
19.3.0.1 | $x^{3} + 4 x + 17$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
1.19.2t1.a.a | $1$ | $ 19 $ | \(\Q(\sqrt{-19}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
1.1463.2t1.a.a | $1$ | $ 7 \cdot 11 \cdot 19 $ | \(\Q(\sqrt{-1463}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
* | 1.77.2t1.a.a | $1$ | $ 7 \cdot 11 $ | \(\Q(\sqrt{77}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
2.1463.4t3.c.a | $2$ | $ 7 \cdot 11 \cdot 19 $ | 4.0.27797.2 | $D_{4}$ (as 4T3) | $1$ | $0$ | |
4.164808413.12t34.b.a | $4$ | $ 7^{3} \cdot 11^{3} \cdot 19^{2}$ | 6.0.528143.1 | $C_3^2:D_4$ (as 6T13) | $1$ | $0$ | |
4.40667011.12t34.b.a | $4$ | $ 7^{2} \cdot 11^{2} \cdot 19^{3}$ | 6.0.528143.1 | $C_3^2:D_4$ (as 6T13) | $1$ | $-2$ | |
4.27797.6t13.b.a | $4$ | $ 7 \cdot 11 \cdot 19^{2}$ | 6.0.528143.1 | $C_3^2:D_4$ (as 6T13) | $1$ | $0$ | |
4.112651.6t13.b.a | $4$ | $ 7^{2} \cdot 11^{2} \cdot 19 $ | 6.0.528143.1 | $C_3^2:D_4$ (as 6T13) | $1$ | $2$ | |
6.152...808.12t201.a.a | $6$ | $ 2^{8} \cdot 7^{3} \cdot 11^{3} \cdot 19^{4}$ | 8.2.170984391424.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $2$ | |
6.289...352.12t202.a.a | $6$ | $ 2^{8} \cdot 7^{3} \cdot 11^{3} \cdot 19^{5}$ | 8.2.170984391424.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $0$ | |
* | 6.2220576512.8t47.a.a | $6$ | $ 2^{8} \cdot 7^{3} \cdot 11^{3} \cdot 19 $ | 8.2.170984391424.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $0$ |
6.42190953728.12t200.a.a | $6$ | $ 2^{8} \cdot 7^{3} \cdot 11^{3} \cdot 19^{2}$ | 8.2.170984391424.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $-2$ | |
9.801628120832.16t1294.a.a | $9$ | $ 2^{8} \cdot 7^{3} \cdot 11^{3} \cdot 19^{3}$ | 8.2.170984391424.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $-1$ | |
9.549...688.18t272.a.a | $9$ | $ 2^{8} \cdot 7^{3} \cdot 11^{3} \cdot 19^{6}$ | 8.2.170984391424.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $1$ | |
9.251...704.18t273.a.a | $9$ | $ 2^{8} \cdot 7^{6} \cdot 11^{6} \cdot 19^{6}$ | 8.2.170984391424.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $1$ | |
9.365...456.18t274.a.a | $9$ | $ 2^{8} \cdot 7^{6} \cdot 11^{6} \cdot 19^{3}$ | 8.2.170984391424.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $-1$ | |
12.122...256.36t1763.a.a | $12$ | $ 2^{16} \cdot 7^{6} \cdot 11^{6} \cdot 19^{7}$ | 8.2.170984391424.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $-2$ | |
12.338...696.24t2821.a.a | $12$ | $ 2^{16} \cdot 7^{6} \cdot 11^{6} \cdot 19^{5}$ | 8.2.170984391424.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $2$ | |
18.201...728.36t1758.a.a | $18$ | $ 2^{16} \cdot 7^{9} \cdot 11^{9} \cdot 19^{9}$ | 8.2.170984391424.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $0$ |