Normalized defining polynomial
\( x^{8} - 4x^{7} + 14x^{5} + 33x^{4} - 94x^{3} - 138x^{2} + 188x + 404 \)
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: | \(4569760000\) \(\medspace = 2^{8}\cdot 5^{4}\cdot 13^{4}\) | 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: | \(16.12\) | 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: | $2\cdot 5^{1/2}13^{1/2}\approx 16.1245154965971$ | ||
Ramified primes: | \(2\), \(5\), \(13\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
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: | \(260=2^{2}\cdot 5\cdot 13\) | ||
Dirichlet character group: | $\lbrace$$\chi_{260}(1,·)$, $\chi_{260}(131,·)$, $\chi_{260}(129,·)$, $\chi_{260}(259,·)$, $\chi_{260}(79,·)$, $\chi_{260}(209,·)$, $\chi_{260}(51,·)$, $\chi_{260}(181,·)$$\rbrace$ | ||
This is a CM field. | |||
Reflex fields: | \(\Q(\sqrt{-1}) \), \(\Q(\sqrt{-5}) \), \(\Q(\sqrt{-13}) \), \(\Q(\sqrt{-65}) \), 8.0.4569760000.1$^{4}$ |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2}a^{4}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{5}-\frac{1}{2}a^{3}$, $\frac{1}{126}a^{6}-\frac{1}{42}a^{5}-\frac{4}{63}a^{4}+\frac{1}{6}a^{3}+\frac{31}{126}a^{2}-\frac{1}{3}a-\frac{10}{63}$, $\frac{1}{42966}a^{7}+\frac{167}{42966}a^{6}-\frac{469}{3069}a^{5}-\frac{3851}{21483}a^{4}-\frac{4841}{42966}a^{3}+\frac{2771}{42966}a^{2}-\frac{6793}{21483}a+\frac{5734}{21483}$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $2$, $3$ |
Class group and class number
$C_{4}$, which has order $4$
Unit group
Rank: | $3$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
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Torsion generator: | \( \frac{13}{2387} a^{7} - \frac{13}{682} a^{6} + \frac{125}{4774} a^{5} - \frac{85}{4774} a^{4} + \frac{645}{4774} a^{3} - \frac{464}{2387} a^{2} + \frac{20}{2387} a + \frac{67}{2387} \) (order $4$) | 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: | $\frac{1}{126}a^{6}-\frac{1}{42}a^{5}-\frac{4}{63}a^{4}+\frac{1}{6}a^{3}+\frac{31}{126}a^{2}-\frac{1}{3}a-\frac{73}{63}$, $\frac{71}{21483}a^{7}-\frac{71}{6138}a^{6}-\frac{419}{42966}a^{5}+\frac{1145}{21483}a^{4}+\frac{7195}{42966}a^{3}-\frac{13331}{42966}a^{2}-\frac{26515}{21483}a-\frac{17812}{21483}$, $\frac{328}{21483}a^{7}-\frac{466}{21483}a^{6}-\frac{1489}{42966}a^{5}-\frac{67}{3069}a^{4}+\frac{25271}{42966}a^{3}+\frac{1820}{3069}a^{2}-\frac{30710}{21483}a-\frac{7609}{3069}$ | 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: | \( 25.5406213931 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{4}\cdot 25.5406213931 \cdot 4}{4\cdot\sqrt{4569760000}}\cr\approx \mathstrut & 0.588849399845 \end{aligned}\]
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 | ${\href{/padicField/3.2.0.1}{2} }^{4}$ | R | ${\href{/padicField/7.2.0.1}{2} }^{4}$ | ${\href{/padicField/11.2.0.1}{2} }^{4}$ | R | ${\href{/padicField/17.2.0.1}{2} }^{4}$ | ${\href{/padicField/19.2.0.1}{2} }^{4}$ | ${\href{/padicField/23.2.0.1}{2} }^{4}$ | ${\href{/padicField/29.1.0.1}{1} }^{8}$ | ${\href{/padicField/31.2.0.1}{2} }^{4}$ | ${\href{/padicField/37.2.0.1}{2} }^{4}$ | ${\href{/padicField/41.2.0.1}{2} }^{4}$ | ${\href{/padicField/43.2.0.1}{2} }^{4}$ | ${\href{/padicField/47.2.0.1}{2} }^{4}$ | ${\href{/padicField/53.2.0.1}{2} }^{4}$ | ${\href{/padicField/59.2.0.1}{2} }^{4}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.4.4.1 | $x^{4} + 6 x^{3} + 17 x^{2} + 24 x + 13$ | $2$ | $2$ | $4$ | $C_2^2$ | $[2]^{2}$ |
2.4.4.1 | $x^{4} + 6 x^{3} + 17 x^{2} + 24 x + 13$ | $2$ | $2$ | $4$ | $C_2^2$ | $[2]^{2}$ | |
\(5\) | 5.4.2.1 | $x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
5.4.2.1 | $x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
\(13\) | 13.4.2.1 | $x^{4} + 284 x^{3} + 21754 x^{2} + 225780 x + 59193$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
13.4.2.1 | $x^{4} + 284 x^{3} + 21754 x^{2} + 225780 x + 59193$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |