Normalized defining polynomial
\( x^{8} - 3x^{6} + 18x^{4} + 4x^{2} + 9 \)
Invariants
Degree: | $8$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[0, 4]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(1235663104\) \(\medspace = 2^{8}\cdot 13^{6}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(13.69\) | 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))
| |
Galois root discriminant: | $2\cdot 13^{3/4}\approx 13.692650084046091$ | ||
Ramified primes: | \(2\), \(13\) | 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)
| |
This field is Galois and abelian over $\Q$. | |||
Conductor: | \(52=2^{2}\cdot 13\) | ||
Dirichlet character group: | $\lbrace$$\chi_{52}(1,·)$, $\chi_{52}(5,·)$, $\chi_{52}(47,·)$, $\chi_{52}(51,·)$, $\chi_{52}(21,·)$, $\chi_{52}(25,·)$, $\chi_{52}(27,·)$, $\chi_{52}(31,·)$$\rbrace$ | ||
This is a CM field. | |||
Reflex fields: | \(\Q(\sqrt{-1}) \), \(\Q(\sqrt{-13}) \), 4.0.2197.1$^{2}$, 8.0.1235663104.1$^{4}$ |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $\frac{1}{3}a^{3}+\frac{1}{3}a$, $\frac{1}{3}a^{4}+\frac{1}{3}a^{2}$, $\frac{1}{3}a^{5}-\frac{1}{3}a$, $\frac{1}{27}a^{6}+\frac{2}{27}a^{4}+\frac{1}{27}a^{2}+\frac{1}{3}$, $\frac{1}{27}a^{7}+\frac{2}{27}a^{5}+\frac{1}{27}a^{3}+\frac{1}{3}a$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $3$ |
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)
| |
Torsion generator: | \( \frac{2}{27} a^{7} - \frac{5}{27} a^{5} + \frac{29}{27} a^{3} + a \) (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)
| |
Fundamental units: | $\frac{2}{27}a^{6}-\frac{5}{27}a^{4}+\frac{20}{27}a^{2}+a+\frac{2}{3}$, $\frac{2}{27}a^{7}-\frac{1}{27}a^{6}-\frac{5}{27}a^{5}+\frac{7}{27}a^{4}+\frac{38}{27}a^{3}-\frac{19}{27}a^{2}+\frac{1}{3}a+\frac{2}{3}$, $\frac{1}{27}a^{6}-\frac{7}{27}a^{4}+\frac{1}{3}a^{3}+\frac{19}{27}a^{2}-\frac{2}{3}a+\frac{1}{3}$ | sage: UK.fundamental_units()
gp: K.fu
magma: [K|fUK(g): g in Generators(UK)];
oscar: [K(fUK(a)) for a in gens(UK)]
| |
Regulator: | \( 35.7224800402 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{4}\cdot 35.7224800402 \cdot 1}{4\cdot\sqrt{1235663104}}\cr\approx \mathstrut & 0.395959753095 \end{aligned}\]
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
\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{13}) \), \(\Q(\sqrt{-13}) \), \(\Q(i, \sqrt{13})\), 4.4.35152.1, 4.0.2197.1 |
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 | R | ${\href{/padicField/3.2.0.1}{2} }^{4}$ | ${\href{/padicField/5.4.0.1}{4} }^{2}$ | ${\href{/padicField/7.4.0.1}{4} }^{2}$ | ${\href{/padicField/11.4.0.1}{4} }^{2}$ | R | ${\href{/padicField/17.2.0.1}{2} }^{4}$ | ${\href{/padicField/19.4.0.1}{4} }^{2}$ | ${\href{/padicField/23.2.0.1}{2} }^{4}$ | ${\href{/padicField/29.1.0.1}{1} }^{8}$ | ${\href{/padicField/31.4.0.1}{4} }^{2}$ | ${\href{/padicField/37.4.0.1}{4} }^{2}$ | ${\href{/padicField/41.4.0.1}{4} }^{2}$ | ${\href{/padicField/43.2.0.1}{2} }^{4}$ | ${\href{/padicField/47.4.0.1}{4} }^{2}$ | ${\href{/padicField/53.1.0.1}{1} }^{8}$ | ${\href{/padicField/59.4.0.1}{4} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.8.8.1 | $x^{8} + 8 x^{7} + 32 x^{6} + 82 x^{5} + 148 x^{4} + 184 x^{3} + 137 x^{2} + 44 x + 5$ | $2$ | $4$ | $8$ | $C_4\times C_2$ | $[2]^{4}$ |
\(13\) | 13.4.3.2 | $x^{4} + 13$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ |
13.4.3.2 | $x^{4} + 13$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ |