Normalized defining polynomial
\( x^{8} - 2x^{7} - 7x^{6} - 2x^{5} + 98x^{4} - 98x^{3} + 67x^{2} - 30x + 9 \)
Invariants
Degree: | $8$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[0, 4]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(4857532416\) \(\medspace = 2^{12}\cdot 3^{4}\cdot 11^{4}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
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Root discriminant: | \(16.25\) | 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))
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Galois root discriminant: | $2^{3/2}3^{1/2}11^{1/2}\approx 16.24807680927192$ | ||
Ramified primes: | \(2\), \(3\), \(11\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
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Discriminant root field: | \(\Q\) | ||
$\card{ \Gal(K/\Q) }$: | $8$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
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This field is Galois and abelian over $\Q$. | |||
Conductor: | \(264=2^{3}\cdot 3\cdot 11\) | ||
Dirichlet character group: | $\lbrace$$\chi_{264}(1,·)$, $\chi_{264}(67,·)$, $\chi_{264}(65,·)$, $\chi_{264}(43,·)$, $\chi_{264}(241,·)$, $\chi_{264}(131,·)$, $\chi_{264}(89,·)$, $\chi_{264}(155,·)$$\rbrace$ | ||
This is a CM field. | |||
Reflex fields: | \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-2}) \), \(\Q(\sqrt{-11}) \), \(\Q(\sqrt{-66}) \), 8.0.4857532416.2$^{4}$ |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{3}a^{5}+\frac{1}{3}a^{3}+\frac{1}{3}a$, $\frac{1}{6}a^{6}-\frac{1}{3}a^{4}-\frac{1}{3}a^{2}-\frac{1}{2}$, $\frac{1}{84870}a^{7}+\frac{5717}{84870}a^{6}-\frac{3862}{42435}a^{5}-\frac{20579}{42435}a^{4}+\frac{9293}{42435}a^{3}+\frac{17998}{42435}a^{2}+\frac{7717}{28290}a+\frac{371}{9430}$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $2$, $3$ |
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)
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Torsion generator: | \( -\frac{754}{14145} a^{7} + \frac{833}{9430} a^{6} + \frac{1862}{4715} a^{5} + \frac{1239}{4715} a^{4} - \frac{23863}{4715} a^{3} + \frac{16807}{4715} a^{2} - \frac{48074}{14145} a + \frac{14281}{9430} \) (order $6$) | 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: | $\frac{17617}{84870}a^{7}-\frac{24301}{84870}a^{6}-\frac{70129}{42435}a^{5}-\frac{60473}{42435}a^{4}+\frac{835106}{42435}a^{3}-\frac{343034}{42435}a^{2}+\frac{68513}{9430}a-\frac{17943}{9430}$, $\frac{9058}{42435}a^{7}-\frac{42953}{84870}a^{6}-\frac{59402}{42435}a^{5}+\frac{10601}{42435}a^{4}+\frac{917623}{42435}a^{3}-\frac{1221097}{42435}a^{2}+\frac{70962}{4715}a-\frac{26129}{9430}$, $\frac{443}{84870}a^{7}+\frac{338}{42435}a^{6}+\frac{679}{42435}a^{5}-\frac{7117}{42435}a^{4}+\frac{14749}{42435}a^{3}-\frac{18811}{42435}a^{2}+\frac{1657}{9430}a-\frac{336}{4715}$ | sage: UK.fundamental_units()
gp: K.fu
magma: [K|fUK(g): g in Generators(UK)];
oscar: [K(fUK(a)) for a in gens(UK)]
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Regulator: | \( 209.789165086 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{4}\cdot 209.789165086 \cdot 2}{6\cdot\sqrt{4857532416}}\cr\approx \mathstrut & 1.56377195285 \end{aligned}\]
Galois group
An abelian group of order 8 |
The 8 conjugacy class representatives for $C_2^3$ |
Character table for $C_2^3$ |
Intermediate fields
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | R | ${\href{/padicField/5.2.0.1}{2} }^{4}$ | ${\href{/padicField/7.2.0.1}{2} }^{4}$ | R | ${\href{/padicField/13.2.0.1}{2} }^{4}$ | ${\href{/padicField/17.2.0.1}{2} }^{4}$ | ${\href{/padicField/19.2.0.1}{2} }^{4}$ | ${\href{/padicField/23.2.0.1}{2} }^{4}$ | ${\href{/padicField/29.2.0.1}{2} }^{4}$ | ${\href{/padicField/31.2.0.1}{2} }^{4}$ | ${\href{/padicField/37.2.0.1}{2} }^{4}$ | ${\href{/padicField/41.2.0.1}{2} }^{4}$ | ${\href{/padicField/43.2.0.1}{2} }^{4}$ | ${\href{/padicField/47.2.0.1}{2} }^{4}$ | ${\href{/padicField/53.2.0.1}{2} }^{4}$ | ${\href{/padicField/59.2.0.1}{2} }^{4}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.4.6.2 | $x^{4} + 4 x^{3} + 16 x^{2} + 24 x + 12$ | $2$ | $2$ | $6$ | $C_2^2$ | $[3]^{2}$ |
2.4.6.2 | $x^{4} + 4 x^{3} + 16 x^{2} + 24 x + 12$ | $2$ | $2$ | $6$ | $C_2^2$ | $[3]^{2}$ | |
\(3\) | 3.2.1.2 | $x^{2} + 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
3.2.1.2 | $x^{2} + 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
3.2.1.2 | $x^{2} + 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
3.2.1.2 | $x^{2} + 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
\(11\) | 11.4.2.1 | $x^{4} + 14 x^{3} + 75 x^{2} + 182 x + 620$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
11.4.2.1 | $x^{4} + 14 x^{3} + 75 x^{2} + 182 x + 620$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |