Normalized defining polynomial
\( x^{8} - x^{7} + 5x^{6} - 9x^{5} + 29x^{4} + 36x^{3} + 80x^{2} + 64x + 256 \)
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: | \(1305015625\) \(\medspace = 5^{6}\cdot 17^{4}\) | 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: | \(13.79\) | 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: | $5^{3/4}17^{1/2}\approx 13.786434567627778$ | ||
Ramified primes: | \(5\), \(17\) | 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: | \(85=5\cdot 17\) | ||
Dirichlet character group: | $\lbrace$$\chi_{85}(1,·)$, $\chi_{85}(67,·)$, $\chi_{85}(69,·)$, $\chi_{85}(33,·)$, $\chi_{85}(16,·)$, $\chi_{85}(18,·)$, $\chi_{85}(52,·)$, $\chi_{85}(84,·)$$\rbrace$ | ||
This is a CM field. | |||
Reflex fields: | \(\Q(\zeta_{5})\)$^{2}$, 4.0.36125.1$^{2}$, 8.0.1305015625.1$^{4}$ |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{116}a^{5}-\frac{1}{4}a^{4}+\frac{1}{4}a^{3}-\frac{1}{4}a^{2}+\frac{1}{4}a+\frac{9}{29}$, $\frac{1}{464}a^{6}-\frac{1}{464}a^{5}-\frac{7}{16}a^{4}+\frac{3}{16}a^{3}+\frac{1}{16}a^{2}+\frac{9}{116}a+\frac{5}{29}$, $\frac{1}{1856}a^{7}-\frac{1}{1856}a^{6}+\frac{5}{1856}a^{5}-\frac{29}{64}a^{4}+\frac{1}{64}a^{3}+\frac{9}{464}a^{2}+\frac{5}{116}a+\frac{1}{29}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
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: | \( -\frac{5}{1856} a^{7} + \frac{25}{1856} a^{6} - \frac{45}{1856} a^{5} + \frac{5}{64} a^{4} - \frac{9}{64} a^{3} + \frac{25}{116} a^{2} + \frac{5}{29} a + \frac{20}{29} \) (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: | $\frac{1}{116}a^{6}+\frac{65}{116}a+1$, $\frac{5}{928}a^{7}-\frac{19}{928}a^{6}+\frac{63}{928}a^{5}-\frac{7}{32}a^{4}+\frac{19}{32}a^{3}-\frac{461}{464}a^{2}+\frac{95}{58}a-\frac{56}{29}$, $\frac{7}{1856}a^{7}-\frac{11}{1856}a^{6}-\frac{9}{1856}a^{5}+\frac{1}{64}a^{4}+\frac{11}{64}a^{3}-\frac{41}{232}a^{2}+\frac{55}{116}a+\frac{4}{29}$ | 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: | \( 35.6323756756 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{4}\cdot 35.6323756756 \cdot 1}{10\cdot\sqrt{1305015625}}\cr\approx \mathstrut & 0.153729210284 \end{aligned}\]
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
\(\Q(\sqrt{85}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{17}) \), \(\Q(\sqrt{5}, \sqrt{17})\), \(\Q(\zeta_{5})\), 4.0.36125.1 |
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 | ${\href{/padicField/7.4.0.1}{4} }^{2}$ | ${\href{/padicField/11.2.0.1}{2} }^{4}$ | ${\href{/padicField/13.4.0.1}{4} }^{2}$ | R | ${\href{/padicField/19.2.0.1}{2} }^{4}$ | ${\href{/padicField/23.4.0.1}{4} }^{2}$ | ${\href{/padicField/29.2.0.1}{2} }^{4}$ | ${\href{/padicField/31.2.0.1}{2} }^{4}$ | ${\href{/padicField/37.4.0.1}{4} }^{2}$ | ${\href{/padicField/41.2.0.1}{2} }^{4}$ | ${\href{/padicField/43.4.0.1}{4} }^{2}$ | ${\href{/padicField/47.4.0.1}{4} }^{2}$ | ${\href{/padicField/53.4.0.1}{4} }^{2}$ | ${\href{/padicField/59.2.0.1}{2} }^{4}$ |
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 | $x^{8} + 16 x^{7} + 104 x^{6} + 352 x^{5} + 674 x^{4} + 784 x^{3} + 776 x^{2} + 928 x + 721$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ |
\(17\) | 17.8.4.1 | $x^{8} + 612 x^{7} + 140536 x^{6} + 14363966 x^{5} + 553913435 x^{4} + 345855654 x^{3} + 4032327212 x^{2} + 6379401496 x + 2294776272$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |