Normalized defining polynomial
\( x^{8} - 2x^{7} - 7x^{6} + 11x^{5} + 20x^{4} - 14x^{3} - 22x^{2} + 3x + 11 \)
Invariants
Degree: | $8$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[0, 4]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(75078125\) \(\medspace = 5^{7}\cdot 31^{2}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(9.65\) | 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: | $5^{7/8}31^{1/2}\approx 22.7656262017263$ | ||
Ramified primes: | \(5\), \(31\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{5}) \) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{451}a^{7}+\frac{182}{451}a^{6}+\frac{107}{451}a^{5}-\frac{145}{451}a^{4}-\frac{51}{451}a^{3}+\frac{73}{451}a^{2}-\frac{120}{451}a+\frac{2}{41}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$
Unit group
Rank: | $3$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
Torsion generator: | \( \frac{59}{451} a^{7} - \frac{86}{451} a^{6} - \frac{452}{451} a^{5} + \frac{465}{451} a^{4} + \frac{1050}{451} a^{3} - \frac{203}{451} a^{2} - \frac{315}{451} a - \frac{5}{41} \) (order $10$) | 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: | $\frac{13}{451}a^{7}+\frac{111}{451}a^{6}-\frac{413}{451}a^{5}-\frac{532}{451}a^{4}+\frac{1592}{451}a^{3}+\frac{1400}{451}a^{2}-\frac{1109}{451}a-\frac{138}{41}$, $\frac{78}{451}a^{7}-\frac{236}{451}a^{6}-\frac{223}{451}a^{5}+\frac{867}{451}a^{4}+\frac{532}{451}a^{3}-\frac{1071}{451}a^{2}-\frac{340}{451}a+\frac{33}{41}$, $\frac{34}{451}a^{7}-\frac{126}{451}a^{6}+\frac{30}{451}a^{5}+\frac{482}{451}a^{4}-\frac{381}{451}a^{3}-\frac{675}{451}a^{2}-\frac{21}{451}a+\frac{27}{41}$ | 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: | \( 18.8167167844 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{4}\cdot 18.8167167844 \cdot 1}{10\cdot\sqrt{75078125}}\cr\approx \mathstrut & 0.338459425309 \end{aligned}\]
Galois group
$\OD_{16}:C_2$ (as 8T16):
A solvable group of order 32 |
The 11 conjugacy class representatives for $(C_8:C_2):C_2$ |
Character table for $(C_8:C_2):C_2$ |
Intermediate fields
\(\Q(\sqrt{5}) \), \(\Q(\zeta_{5})\) |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Galois closure: | data not computed |
Degree 8 sibling: | data not computed |
Degree 16 siblings: | 16.4.5205633773443603515625.1, 16.0.5205633773443603515625.1, 16.0.5416892584228515625.1 |
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 | ${\href{/padicField/2.8.0.1}{8} }$ | ${\href{/padicField/3.8.0.1}{8} }$ | R | ${\href{/padicField/7.8.0.1}{8} }$ | ${\href{/padicField/11.2.0.1}{2} }^{2}{,}\,{\href{/padicField/11.1.0.1}{1} }^{4}$ | ${\href{/padicField/13.8.0.1}{8} }$ | ${\href{/padicField/17.8.0.1}{8} }$ | ${\href{/padicField/19.4.0.1}{4} }^{2}$ | ${\href{/padicField/23.8.0.1}{8} }$ | ${\href{/padicField/29.2.0.1}{2} }^{4}$ | R | ${\href{/padicField/37.8.0.1}{8} }$ | ${\href{/padicField/41.1.0.1}{1} }^{8}$ | ${\href{/padicField/43.8.0.1}{8} }$ | ${\href{/padicField/47.8.0.1}{8} }$ | ${\href{/padicField/53.8.0.1}{8} }$ | ${\href{/padicField/59.4.0.1}{4} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(5\) | 5.8.7.2 | $x^{8} + 5$ | $8$ | $1$ | $7$ | $C_8:C_2$ | $[\ ]_{8}^{2}$ |
\(31\) | $\Q_{31}$ | $x + 28$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{31}$ | $x + 28$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
31.2.1.2 | $x^{2} + 31$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
31.2.0.1 | $x^{2} + 29 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
31.2.1.1 | $x^{2} + 93$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
1.155.2t1.a.a | $1$ | $ 5 \cdot 31 $ | \(\Q(\sqrt{-155}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
* | 1.5.2t1.a.a | $1$ | $ 5 $ | \(\Q(\sqrt{5}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
1.31.2t1.a.a | $1$ | $ 31 $ | \(\Q(\sqrt{-31}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
* | 1.5.4t1.a.a | $1$ | $ 5 $ | \(\Q(\zeta_{5})\) | $C_4$ (as 4T1) | $0$ | $-1$ |
1.155.4t1.a.a | $1$ | $ 5 \cdot 31 $ | 4.4.120125.1 | $C_4$ (as 4T1) | $0$ | $1$ | |
* | 1.5.4t1.a.b | $1$ | $ 5 $ | \(\Q(\zeta_{5})\) | $C_4$ (as 4T1) | $0$ | $-1$ |
1.155.4t1.a.b | $1$ | $ 5 \cdot 31 $ | 4.4.120125.1 | $C_4$ (as 4T1) | $0$ | $1$ | |
2.155.4t3.b.a | $2$ | $ 5 \cdot 31 $ | 4.0.4805.1 | $D_{4}$ (as 4T3) | $1$ | $0$ | |
2.775.4t3.c.a | $2$ | $ 5^{2} \cdot 31 $ | 4.2.3875.1 | $D_{4}$ (as 4T3) | $1$ | $0$ | |
* | 4.600625.8t16.a.a | $4$ | $ 5^{4} \cdot 31^{2}$ | 8.0.75078125.1 | $(C_8:C_2):C_2$ (as 8T16) | $1$ | $0$ |