Normalized defining polynomial
\( x^{8} - 2x^{7} + x^{6} - 14x^{5} + 96x^{4} - 82x^{3} - 17x^{2} + 86x + 31 \)
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$$\chi_{264}(1,·)$, $\chi_{264}(133,·)$, $\chi_{264}(65,·)$, $\chi_{264}(109,·)$, $\chi_{264}(241,·)$, $\chi_{264}(89,·)$, $\chi_{264}(221,·)$, $\chi_{264}(197,·)$$\rbrace$ | ||
This is a CM field. | |||
Reflex fields: | \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-11}) \), \(\Q(\sqrt{-6}) \), \(\Q(\sqrt{-22}) \), 8.0.4857532416.1$^{4}$ |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{22}a^{6}+\frac{4}{11}a^{5}-\frac{5}{11}a^{4}-\frac{3}{11}a^{3}+\frac{1}{11}a+\frac{3}{22}$, $\frac{1}{245410}a^{7}-\frac{2021}{245410}a^{6}-\frac{9157}{24541}a^{5}+\frac{3913}{11155}a^{4}-\frac{28629}{122705}a^{3}+\frac{1571}{24541}a^{2}-\frac{60617}{245410}a-\frac{73781}{245410}$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $2$ |
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)
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Torsion generator: | \( -\frac{122}{1265} a^{7} + \frac{579}{2530} a^{6} - \frac{50}{253} a^{5} + \frac{1838}{1265} a^{4} - \frac{12389}{1265} a^{3} + \frac{3007}{253} a^{2} - \frac{5436}{1265} a - \frac{13681}{2530} \) (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: | $\frac{5286}{24541}a^{7}-\frac{26497}{49082}a^{6}+\frac{12126}{24541}a^{5}-\frac{80097}{24541}a^{4}+\frac{547191}{24541}a^{3}-\frac{715373}{24541}a^{2}+\frac{269171}{24541}a+\frac{652095}{49082}$, $\frac{41026}{122705}a^{7}-\frac{208807}{245410}a^{6}+\frac{19402}{24541}a^{5}-\frac{622684}{122705}a^{4}+\frac{4270177}{122705}a^{3}-\frac{1139067}{24541}a^{2}+\frac{2409828}{122705}a+\frac{4457663}{245410}$, $\frac{13446}{122705}a^{7}-\frac{23096}{122705}a^{6}-\frac{108}{2231}a^{5}-\frac{175854}{122705}a^{4}+\frac{1231772}{122705}a^{3}-\frac{110434}{24541}a^{2}-\frac{132282}{11155}a+\frac{969929}{122705}$ | 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: | \( 131.384706296 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{4}\cdot 131.384706296 \cdot 1}{6\cdot\sqrt{4857532416}}\cr\approx \mathstrut & 0.489671901441 \end{aligned}\]
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 | ${\href{/padicField/13.2.0.1}{2} }^{4}$ | ${\href{/padicField/17.2.0.1}{2} }^{4}$ | ${\href{/padicField/19.2.0.1}{2} }^{4}$ | ${\href{/padicField/23.2.0.1}{2} }^{4}$ | ${\href{/padicField/29.2.0.1}{2} }^{4}$ | ${\href{/padicField/31.1.0.1}{1} }^{8}$ | ${\href{/padicField/37.2.0.1}{2} }^{4}$ | ${\href{/padicField/41.2.0.1}{2} }^{4}$ | ${\href{/padicField/43.2.0.1}{2} }^{4}$ | ${\href{/padicField/47.2.0.1}{2} }^{4}$ | ${\href{/padicField/53.2.0.1}{2} }^{4}$ | ${\href{/padicField/59.2.0.1}{2} }^{4}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.4.6.1 | $x^{4} + 2 x^{3} + 31 x^{2} + 30 x + 183$ | $2$ | $2$ | $6$ | $C_2^2$ | $[3]^{2}$ |
2.4.6.1 | $x^{4} + 2 x^{3} + 31 x^{2} + 30 x + 183$ | $2$ | $2$ | $6$ | $C_2^2$ | $[3]^{2}$ | |
\(3\) | 3.4.2.1 | $x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
3.4.2.1 | $x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
\(11\) | 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 | $x^{4} + 14 x^{3} + 75 x^{2} + 182 x + 620$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |