Normalized defining polynomial
\( x^{8} - 2x^{7} + 23x^{6} - 29x^{5} + 220x^{4} - 124x^{3} + 1013x^{2} - 142x + 1801 \)
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: | \(6081328125\) \(\medspace = 3^{4}\cdot 5^{7}\cdot 31^{2}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(16.71\) | 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: | $3^{1/2}5^{7/8}31^{1/2}\approx 39.43122124751123$ | ||
Ramified primes: | \(3\), \(5\), \(31\) | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{3}a^{4}-\frac{1}{3}a^{3}-\frac{1}{3}a^{2}+\frac{1}{3}a+\frac{1}{3}$, $\frac{1}{3}a^{5}+\frac{1}{3}a^{3}-\frac{1}{3}a+\frac{1}{3}$, $\frac{1}{3}a^{6}+\frac{1}{3}a^{3}-\frac{1}{3}$, $\frac{1}{101613}a^{7}-\frac{1372}{33871}a^{6}-\frac{751}{33871}a^{5}-\frac{3947}{33871}a^{4}+\frac{41467}{101613}a^{3}+\frac{12865}{101613}a^{2}+\frac{14776}{101613}a+\frac{9839}{101613}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{2}$, which has order $2$
Unit group
Rank: | $3$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
Torsion generator: | \( \frac{1910}{101613} a^{7} - \frac{3488}{101613} a^{6} + \frac{32258}{101613} a^{5} - \frac{24353}{101613} a^{4} + \frac{214798}{101613} a^{3} + \frac{49546}{101613} a^{2} + \frac{549553}{101613} a + \frac{122222}{33871} \) (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: | $\frac{971}{101613}a^{7}+\frac{142}{101613}a^{6}+\frac{13952}{101613}a^{5}+\frac{18529}{101613}a^{4}+\frac{93451}{101613}a^{3}+\frac{264484}{101613}a^{2}+\frac{291031}{101613}a+\frac{328102}{33871}$, $\frac{3053}{101613}a^{7}-\frac{7}{101613}a^{6}+\frac{10425}{33871}a^{5}+\frac{91397}{101613}a^{4}+\frac{90566}{101613}a^{3}+\frac{334205}{33871}a^{2}+\frac{62698}{101613}a+\frac{2704570}{101613}$, $\frac{4054}{101613}a^{7}+\frac{12139}{101613}a^{6}+\frac{3836}{33871}a^{5}+\frac{398245}{101613}a^{4}-\frac{265523}{101613}a^{3}+\frac{1183275}{33871}a^{2}-\frac{1032025}{101613}a+\frac{9301793}{101613}$ | 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: | \( 77.0442623514 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{4}\cdot 77.0442623514 \cdot 2}{10\cdot\sqrt{6081328125}}\cr\approx \mathstrut & 0.307957334577 \end{aligned}\]
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: | 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 | ${\href{/padicField/2.8.0.1}{8} }$ | R | 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 | ${\href{/padicField/37.8.0.1}{8} }$ | ${\href{/padicField/41.2.0.1}{2} }^{4}$ | ${\href{/padicField/43.8.0.1}{8} }$ | ${\href{/padicField/47.8.0.1}{8} }$ | ${\href{/padicField/53.8.0.1}{8} }$ | ${\href{/padicField/59.4.0.1}{4} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(3\) | 3.8.4.2 | $x^{8} - 6 x^{6} + 162$ | $2$ | $4$ | $4$ | $C_8$ | $[\ ]_{2}^{4}$ |
\(5\) | 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 | $x^{2} + 31$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
31.2.1.1 | $x^{2} + 93$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
31.2.0.1 | $x^{2} + 29 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
1.31.2t1.a.a | $1$ | $ 31 $ | \(\Q(\sqrt{-31}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
* | 1.5.2t1.a.a | $1$ | $ 5 $ | \(\Q(\sqrt{5}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
1.155.2t1.a.a | $1$ | $ 5 \cdot 31 $ | \(\Q(\sqrt{-155}) \) | $C_2$ (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) | $0$ | $1$ | |
2.775.4t3.b.a | $2$ | $ 5^{2} \cdot 31 $ | 4.0.120125.1 | $D_{4}$ (as 4T3) | $1$ | $0$ | |
2.155.4t3.c.a | $2$ | $ 5 \cdot 31 $ | 4.2.775.1 | $D_{4}$ (as 4T3) | $1$ | $0$ | |
* | 4.48650625.8t16.a.a | $4$ | $ 3^{4} \cdot 5^{4} \cdot 31^{2}$ | 8.0.6081328125.1 | $(C_8:C_2):C_2$ (as 8T16) | $1$ | $0$ |