Normalized defining polynomial
\( x^{8} - 30x^{6} + 193x^{4} - 424x^{2} + 256 \)
Invariants
Degree: | $8$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[8, 0]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(667841990656\) \(\medspace = 2^{12}\cdot 113^{4}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(30.07\) | 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: | $2^{3/2}113^{1/2}\approx 30.066592756745816$ | ||
Ramified primes: | \(2\), \(113\) | 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 over $\Q$. | |||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{4}a^{4}-\frac{1}{4}a^{2}-\frac{1}{2}a$, $\frac{1}{4}a^{5}-\frac{1}{4}a^{3}-\frac{1}{2}a^{2}$, $\frac{1}{8}a^{6}+\frac{3}{8}a^{2}-\frac{1}{2}a$, $\frac{1}{32}a^{7}+\frac{1}{16}a^{5}+\frac{1}{32}a^{3}-\frac{1}{4}a$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $2$ |
Class group and class number
$C_{2}$, which has order $2$
Unit group
Rank: | $7$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
Torsion generator: | \( -1 \) (order $2$) | 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: | $\frac{9}{32}a^{7}-\frac{119}{16}a^{5}+\frac{889}{32}a^{3}-\frac{75}{4}a-1$, $\frac{3}{32}a^{7}+\frac{1}{4}a^{6}-\frac{41}{16}a^{5}-\frac{27}{4}a^{4}+\frac{363}{32}a^{3}+28a^{2}-\frac{51}{4}a-27$, $\frac{3}{32}a^{7}-\frac{1}{4}a^{6}-\frac{41}{16}a^{5}+\frac{27}{4}a^{4}+\frac{363}{32}a^{3}-28a^{2}-\frac{51}{4}a+27$, $\frac{3}{32}a^{7}+\frac{1}{2}a^{6}-\frac{41}{16}a^{5}-\frac{53}{4}a^{4}+\frac{363}{32}a^{3}+\frac{201}{4}a^{2}-\frac{47}{4}a-39$, $\frac{3}{32}a^{7}-\frac{1}{2}a^{6}-\frac{41}{16}a^{5}+\frac{53}{4}a^{4}+\frac{363}{32}a^{3}-\frac{201}{4}a^{2}-\frac{47}{4}a+39$, $\frac{1}{8}a^{7}+\frac{1}{4}a^{6}-\frac{13}{4}a^{5}-\frac{13}{2}a^{4}+\frac{85}{8}a^{3}+\frac{93}{4}a^{2}-3a-13$, $a^{7}-2a^{6}-\frac{53}{2}a^{5}+53a^{4}+100a^{3}-200a^{2}-\frac{135}{2}a+135$ | 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: | \( 8072.46389055 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{8}\cdot(2\pi)^{0}\cdot 8072.46389055 \cdot 2}{2\cdot\sqrt{667841990656}}\cr\approx \mathstrut & 2.52876932902 \end{aligned}\]
Galois group
A solvable group of order 8 |
The 5 conjugacy class representatives for $D_4$ |
Character table for $D_4$ |
Intermediate fields
\(\Q(\sqrt{226}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{113}) \), \(\Q(\sqrt{2}, \sqrt{113})\), 4.4.7232.1 x2, 4.4.102152.1 x2 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 4 siblings: | 4.4.102152.1, 4.4.7232.1 |
Minimal sibling: | 4.4.7232.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | ${\href{/padicField/3.4.0.1}{4} }^{2}$ | ${\href{/padicField/5.4.0.1}{4} }^{2}$ | ${\href{/padicField/7.2.0.1}{2} }^{4}$ | ${\href{/padicField/11.2.0.1}{2} }^{4}$ | ${\href{/padicField/13.2.0.1}{2} }^{4}$ | ${\href{/padicField/17.2.0.1}{2} }^{4}$ | ${\href{/padicField/19.4.0.1}{4} }^{2}$ | ${\href{/padicField/23.2.0.1}{2} }^{4}$ | ${\href{/padicField/29.4.0.1}{4} }^{2}$ | ${\href{/padicField/31.2.0.1}{2} }^{4}$ | ${\href{/padicField/37.4.0.1}{4} }^{2}$ | ${\href{/padicField/41.1.0.1}{1} }^{8}$ | ${\href{/padicField/43.4.0.1}{4} }^{2}$ | ${\href{/padicField/47.2.0.1}{2} }^{4}$ | ${\href{/padicField/53.2.0.1}{2} }^{4}$ | ${\href{/padicField/59.4.0.1}{4} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.2.3.1 | $x^{2} + 4 x + 2$ | $2$ | $1$ | $3$ | $C_2$ | $[3]$ |
2.2.3.1 | $x^{2} + 4 x + 2$ | $2$ | $1$ | $3$ | $C_2$ | $[3]$ | |
2.2.3.1 | $x^{2} + 4 x + 2$ | $2$ | $1$ | $3$ | $C_2$ | $[3]$ | |
2.2.3.1 | $x^{2} + 4 x + 2$ | $2$ | $1$ | $3$ | $C_2$ | $[3]$ | |
\(113\) | 113.2.1.1 | $x^{2} + 113$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
113.2.1.1 | $x^{2} + 113$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
113.2.1.1 | $x^{2} + 113$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
113.2.1.1 | $x^{2} + 113$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
* | 1.113.2t1.a.a | $1$ | $ 113 $ | \(\Q(\sqrt{113}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
* | 1.8.2t1.a.a | $1$ | $ 2^{3}$ | \(\Q(\sqrt{2}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
* | 1.904.2t1.b.a | $1$ | $ 2^{3} \cdot 113 $ | \(\Q(\sqrt{226}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
*2 | 2.904.4t3.c.a | $2$ | $ 2^{3} \cdot 113 $ | 8.8.667841990656.1 | $D_4$ (as 8T4) | $1$ | $2$ |