Normalized defining polynomial
\( x^{8} - 2x^{7} + 2x^{6} - 5x^{5} + 10x^{4} + 11x^{3} + 20x^{2} + 11x + 1 \)
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: | \(-3125131875\) \(\medspace = -\,3^{6}\cdot 5^{4}\cdot 19^{3}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(15.38\) | 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: | $3^{3/4}5^{1/2}19^{1/2}\approx 22.21788649167895$ | ||
Ramified primes: | \(3\), \(5\), \(19\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{-19}) \) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{7}a^{6}-\frac{3}{7}a^{5}+\frac{3}{7}a^{3}+\frac{3}{7}a+\frac{3}{7}$, $\frac{1}{49}a^{7}-\frac{1}{49}a^{6}+\frac{8}{49}a^{5}-\frac{18}{49}a^{4}-\frac{8}{49}a^{3}+\frac{24}{49}a^{2}-\frac{5}{49}a-\frac{22}{49}$
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: | $\frac{5}{49}a^{7}-\frac{5}{49}a^{6}-\frac{9}{49}a^{5}+\frac{8}{49}a^{4}+\frac{9}{49}a^{3}+\frac{120}{49}a^{2}+\frac{73}{49}a+\frac{37}{49}$, $\frac{5}{49}a^{7}-\frac{19}{49}a^{6}+\frac{33}{49}a^{5}-\frac{41}{49}a^{4}+\frac{65}{49}a^{3}-\frac{27}{49}a^{2}+\frac{31}{49}a-\frac{5}{49}$, $\frac{5}{49}a^{7}-\frac{19}{49}a^{6}+\frac{33}{49}a^{5}-\frac{41}{49}a^{4}+\frac{65}{49}a^{3}-\frac{27}{49}a^{2}+\frac{31}{49}a+\frac{44}{49}$, $\frac{81}{49}a^{7}-\frac{263}{49}a^{6}+\frac{361}{49}a^{5}-\frac{478}{49}a^{4}+\frac{962}{49}a^{3}+\frac{278}{49}a^{2}-\frac{20}{49}a-\frac{25}{49}$ | 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: | \( 40.4800899251 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{3}\cdot 40.4800899251 \cdot 1}{2\cdot\sqrt{3125131875}}\cr\approx \mathstrut & 0.359233554140 \end{aligned}\]
Galois group
A solvable group of order 32 |
The 11 conjugacy class representatives for $Z_8 : Z_8^\times$ |
Character table for $Z_8 : Z_8^\times$ |
Intermediate fields
\(\Q(\sqrt{5}) \), 4.2.4275.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | ${\href{/padicField/2.8.0.1}{8} }$ | R | R | ${\href{/padicField/7.2.0.1}{2} }^{4}$ | ${\href{/padicField/11.2.0.1}{2} }^{2}{,}\,{\href{/padicField/11.1.0.1}{1} }^{4}$ | ${\href{/padicField/13.8.0.1}{8} }$ | ${\href{/padicField/17.4.0.1}{4} }^{2}$ | R | ${\href{/padicField/23.4.0.1}{4} }^{2}$ | ${\href{/padicField/29.2.0.1}{2} }^{3}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.2.0.1}{2} }^{3}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.8.0.1}{8} }$ | ${\href{/padicField/41.2.0.1}{2} }^{3}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.2.0.1}{2} }^{4}$ | ${\href{/padicField/47.4.0.1}{4} }^{2}$ | ${\href{/padicField/53.8.0.1}{8} }$ | ${\href{/padicField/59.2.0.1}{2} }^{3}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(3\) | 3.8.6.3 | $x^{8} - 6 x^{4} + 18$ | $4$ | $2$ | $6$ | $C_8:C_2$ | $[\ ]_{4}^{4}$ |
\(5\) | 5.8.4.1 | $x^{8} + 80 x^{7} + 2428 x^{6} + 33688 x^{5} + 195810 x^{4} + 305952 x^{3} + 870132 x^{2} + 1037416 x + 503089$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |
\(19\) | $\Q_{19}$ | $x + 17$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{19}$ | $x + 17$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
19.2.1.2 | $x^{2} + 19$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
19.2.1.2 | $x^{2} + 19$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
19.2.1.2 | $x^{2} + 19$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
1.57.2t1.a.a | $1$ | $ 3 \cdot 19 $ | \(\Q(\sqrt{57}) \) | $C_2$ (as 2T1) | $1$ | $1$ | |
1.19.2t1.a.a | $1$ | $ 19 $ | \(\Q(\sqrt{-19}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
1.3.2t1.a.a | $1$ | $ 3 $ | \(\Q(\sqrt{-3}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
* | 1.5.2t1.a.a | $1$ | $ 5 $ | \(\Q(\sqrt{5}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
1.285.2t1.a.a | $1$ | $ 3 \cdot 5 \cdot 19 $ | \(\Q(\sqrt{285}) \) | $C_2$ (as 2T1) | $1$ | $1$ | |
1.95.2t1.a.a | $1$ | $ 5 \cdot 19 $ | \(\Q(\sqrt{-95}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
1.15.2t1.a.a | $1$ | $ 3 \cdot 5 $ | \(\Q(\sqrt{-15}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
* | 2.855.4t3.b.a | $2$ | $ 3^{2} \cdot 5 \cdot 19 $ | 4.2.4275.1 | $D_{4}$ (as 4T3) | $1$ | $0$ |
2.95.4t3.c.a | $2$ | $ 5 \cdot 19 $ | 4.2.475.1 | $D_{4}$ (as 4T3) | $1$ | $0$ | |
* | 4.731025.8t15.b.a | $4$ | $ 3^{4} \cdot 5^{2} \cdot 19^{2}$ | 8.2.3125131875.2 | $Z_8 : Z_8^\times$ (as 8T15) | $1$ | $0$ |