Normalized defining polynomial
\( x^{8} - 2x^{7} - x^{6} - 60x^{5} + 214x^{4} - 89x^{3} + 557x^{2} - 2910x + 2601 \)
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)
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Discriminant: | \(-2884800683080019\) \(\medspace = -\,11^{7}\cdot 23^{6}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(85.61\) | 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: | $11^{7/8}23^{3/4}\approx 85.60802473392998$ | ||
Ramified primes: | \(11\), \(23\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{-11}) \) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{3}a^{5}+\frac{1}{3}a$, $\frac{1}{3}a^{6}+\frac{1}{3}a^{2}$, $\frac{1}{2446779}a^{7}+\frac{53020}{2446779}a^{6}-\frac{122632}{2446779}a^{5}-\frac{367387}{815593}a^{4}+\frac{89308}{2446779}a^{3}+\frac{771322}{2446779}a^{2}-\frac{875344}{2446779}a+\frac{152791}{815593}$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $3$ |
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)
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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)
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Fundamental units: | $\frac{22225}{2446779}a^{7}-\frac{162385}{2446779}a^{6}-\frac{471860}{815593}a^{5}+\frac{2987820}{815593}a^{4}-\frac{9254585}{2446779}a^{3}+\frac{11100485}{2446779}a^{2}-\frac{47092720}{815593}a+\frac{81210023}{815593}$, $\frac{25861}{2446779}a^{7}+\frac{46129}{815593}a^{6}+\frac{151675}{815593}a^{5}-\frac{152350}{815593}a^{4}-\frac{5058746}{2446779}a^{3}-\frac{1564439}{815593}a^{2}+\frac{3643611}{815593}a+\frac{20169791}{815593}$, $\frac{37821}{815593}a^{7}-\frac{5708}{2446779}a^{6}-\frac{178036}{2446779}a^{5}-\frac{1504137}{815593}a^{4}+\frac{1978441}{815593}a^{3}+\frac{8272744}{2446779}a^{2}+\frac{48616883}{2446779}a-\frac{24587365}{815593}$, $\frac{368443}{2446779}a^{7}+\frac{579917}{2446779}a^{6}+\frac{950224}{2446779}a^{5}-\frac{6985347}{815593}a^{4}-\frac{1823327}{2446779}a^{3}-\frac{4373611}{2446779}a^{2}+\frac{232092568}{2446779}a-\frac{55361550}{815593}$ | 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: | \( 47306.1646777 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{3}\cdot 47306.1646777 \cdot 1}{2\cdot\sqrt{2884800683080019}}\cr\approx \mathstrut & 0.436947676112 \end{aligned}\]
Galois group
$\SD_{16}$ (as 8T8):
A solvable group of order 16 |
The 7 conjugacy class representatives for $QD_{16}$ |
Character table for $QD_{16}$ |
Intermediate fields
\(\Q(\sqrt{253}) \), 4.2.704099.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Galois closure: | 16.0.8322074981098944220712357040361.1 |
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 | ${\href{/padicField/2.8.0.1}{8} }$ | ${\href{/padicField/3.2.0.1}{2} }^{4}$ | ${\href{/padicField/5.4.0.1}{4} }^{2}$ | ${\href{/padicField/7.2.0.1}{2} }^{3}{,}\,{\href{/padicField/7.1.0.1}{1} }^{2}$ | R | ${\href{/padicField/13.8.0.1}{8} }$ | ${\href{/padicField/17.2.0.1}{2} }^{3}{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ | ${\href{/padicField/19.2.0.1}{2} }^{3}{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | R | ${\href{/padicField/29.8.0.1}{8} }$ | ${\href{/padicField/31.4.0.1}{4} }^{2}$ | ${\href{/padicField/37.4.0.1}{4} }^{2}$ | ${\href{/padicField/41.8.0.1}{8} }$ | ${\href{/padicField/43.2.0.1}{2} }^{3}{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | ${\href{/padicField/47.2.0.1}{2} }^{4}$ | ${\href{/padicField/53.4.0.1}{4} }^{2}$ | ${\href{/padicField/59.4.0.1}{4} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(11\) | 11.8.7.2 | $x^{8} + 22$ | $8$ | $1$ | $7$ | $QD_{16}$ | $[\ ]_{8}^{2}$ |
\(23\) | 23.8.6.1 | $x^{8} - 138 x^{4} - 217948$ | $4$ | $2$ | $6$ | $Q_8$ | $[\ ]_{4}^{2}$ |