Normalized defining polynomial
\( x^{8} - x^{7} - 5x^{6} + 11x^{5} + 19x^{4} + 66x^{3} - 180x^{2} - 216x + 1296 \)
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: | \(4372515625\) \(\medspace = 5^{6}\cdot 23^{4}\) | 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: | \(16.04\) | 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: | $5^{3/4}23^{1/2}\approx 16.03582917757843$ | ||
Ramified primes: | \(5\), \(23\) | 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: | \(115=5\cdot 23\) | ||
Dirichlet character group: | $\lbrace$$\chi_{115}(1,·)$, $\chi_{115}(68,·)$, $\chi_{115}(47,·)$, $\chi_{115}(114,·)$, $\chi_{115}(22,·)$, $\chi_{115}(24,·)$, $\chi_{115}(91,·)$, $\chi_{115}(93,·)$$\rbrace$ | ||
This is a CM field. | |||
Reflex fields: | \(\Q(\sqrt{-23}) \), \(\Q(\sqrt{-115}) \), \(\Q(\zeta_{5})\)$^{2}$, 8.0.4372515625.1$^{4}$ |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{114}a^{5}-\frac{1}{6}a^{4}+\frac{1}{6}a^{3}-\frac{1}{6}a^{2}+\frac{1}{6}a-\frac{8}{19}$, $\frac{1}{684}a^{6}-\frac{1}{684}a^{5}+\frac{13}{36}a^{4}-\frac{7}{36}a^{3}+\frac{1}{36}a^{2}+\frac{11}{114}a-\frac{5}{19}$, $\frac{1}{4104}a^{7}-\frac{1}{4104}a^{6}-\frac{5}{4104}a^{5}-\frac{79}{216}a^{4}+\frac{1}{216}a^{3}+\frac{11}{684}a^{2}-\frac{5}{114}a-\frac{1}{19}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{3}$, which has order $3$
Unit group
Rank: | $3$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
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Torsion generator: | \( -\frac{5}{4104} a^{7} - \frac{25}{4104} a^{6} + \frac{55}{4104} a^{5} + \frac{5}{216} a^{4} - \frac{11}{216} a^{3} - \frac{25}{114} a^{2} - \frac{5}{19} a + \frac{30}{19} \) (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)
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Fundamental units: | $\frac{5}{4104}a^{7}+\frac{25}{4104}a^{6}-\frac{55}{4104}a^{5}-\frac{5}{216}a^{4}+\frac{11}{216}a^{3}+\frac{25}{114}a^{2}+\frac{5}{19}a-\frac{11}{19}$, $\frac{1}{4104}a^{7}+\frac{11}{4104}a^{6}+\frac{55}{4104}a^{5}+\frac{5}{216}a^{4}-\frac{11}{216}a^{3}-\frac{179}{684}a^{2}-\frac{59}{114}a-\frac{8}{19}$, $\frac{17}{684}a^{6}-\frac{11}{684}a^{5}-\frac{1}{36}a^{4}-\frac{5}{36}a^{3}+\frac{11}{36}a^{2}+\frac{103}{57}a-\frac{17}{19}$ | 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: | \( 39.3022218671 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{4}\cdot 39.3022218671 \cdot 3}{10\cdot\sqrt{4372515625}}\cr\approx \mathstrut & 0.277902303167 \end{aligned}\]
Galois group
$C_2\times C_4$ (as 8T2):
An abelian group of order 8 |
The 8 conjugacy class representatives for $C_4\times C_2$ |
Character table for $C_4\times C_2$ |
Intermediate fields
\(\Q(\sqrt{-115}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-23}) \), \(\Q(\sqrt{5}, \sqrt{-23})\), \(\Q(\zeta_{5})\), 4.4.66125.1 |
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 | ${\href{/padicField/2.4.0.1}{4} }^{2}$ | ${\href{/padicField/3.4.0.1}{4} }^{2}$ | R | ${\href{/padicField/7.4.0.1}{4} }^{2}$ | ${\href{/padicField/11.2.0.1}{2} }^{4}$ | ${\href{/padicField/13.4.0.1}{4} }^{2}$ | ${\href{/padicField/17.4.0.1}{4} }^{2}$ | ${\href{/padicField/19.2.0.1}{2} }^{4}$ | R | ${\href{/padicField/29.2.0.1}{2} }^{4}$ | ${\href{/padicField/31.1.0.1}{1} }^{8}$ | ${\href{/padicField/37.4.0.1}{4} }^{2}$ | ${\href{/padicField/41.1.0.1}{1} }^{8}$ | ${\href{/padicField/43.4.0.1}{4} }^{2}$ | ${\href{/padicField/47.4.0.1}{4} }^{2}$ | ${\href{/padicField/53.4.0.1}{4} }^{2}$ | ${\href{/padicField/59.2.0.1}{2} }^{4}$ |
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 |
---|---|---|---|---|---|---|---|
\(5\) | 5.8.6.1 | $x^{8} + 16 x^{7} + 104 x^{6} + 352 x^{5} + 674 x^{4} + 784 x^{3} + 776 x^{2} + 928 x + 721$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ |
\(23\) | 23.8.4.1 | $x^{8} + 98 x^{6} + 38 x^{5} + 3331 x^{4} - 1634 x^{3} + 44919 x^{2} - 57494 x + 224528$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |