Normalized defining polynomial
\( x^{8} + 40x^{6} + 500x^{4} + 2000x^{2} + 2450 \)
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: | \(33554432000000\) \(\medspace = 2^{31}\cdot 5^{6}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(49.06\) | 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: | $2^{31/8}5^{3/4}\approx 49.05900508138431$ | ||
Ramified primes: | \(2\), \(5\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{2}) \) | ||
$\card{ \Gal(K/\Q) }$: | $8$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
This field is Galois and abelian over $\Q$. | |||
Conductor: | \(160=2^{5}\cdot 5\) | ||
Dirichlet character group: | $\lbrace$$\chi_{160}(1,·)$, $\chi_{160}(133,·)$, $\chi_{160}(9,·)$, $\chi_{160}(77,·)$, $\chi_{160}(81,·)$, $\chi_{160}(53,·)$, $\chi_{160}(89,·)$, $\chi_{160}(157,·)$$\rbrace$ | ||
This is a CM field. | |||
Reflex fields: | 8.0.33554432000000.2$^{8}$ |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{5}a^{4}$, $\frac{1}{35}a^{5}-\frac{1}{7}a^{3}-\frac{2}{7}a$, $\frac{1}{35}a^{6}+\frac{2}{35}a^{4}-\frac{2}{7}a^{2}$, $\frac{1}{35}a^{7}-\frac{3}{7}a$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $7$ |
Class group and class number
$C_{164}$, which has order $164$
Unit group
Rank: | $3$ | 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: | $\frac{1}{5}a^{4}+4a^{2}+9$, $\frac{1}{35}a^{6}+\frac{44}{35}a^{4}+\frac{103}{7}a^{2}+29$, $\frac{2}{35}a^{6}+\frac{46}{35}a^{4}+\frac{45}{7}a^{2}+9$ | 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: | \( 78.6242549932 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{4}\cdot 78.6242549932 \cdot 164}{2\cdot\sqrt{33554432000000}}\cr\approx \mathstrut & 1.73466223368 \end{aligned}\]
Galois group
A cyclic group of order 8 |
The 8 conjugacy class representatives for $C_8$ |
Character table for $C_8$ |
Intermediate fields
\(\Q(\sqrt{2}) \), 4.4.51200.1 |
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 | ${\href{/padicField/3.8.0.1}{8} }$ | R | ${\href{/padicField/7.1.0.1}{1} }^{8}$ | ${\href{/padicField/11.8.0.1}{8} }$ | ${\href{/padicField/13.8.0.1}{8} }$ | ${\href{/padicField/17.4.0.1}{4} }^{2}$ | ${\href{/padicField/19.8.0.1}{8} }$ | ${\href{/padicField/23.1.0.1}{1} }^{8}$ | ${\href{/padicField/29.8.0.1}{8} }$ | ${\href{/padicField/31.1.0.1}{1} }^{8}$ | ${\href{/padicField/37.8.0.1}{8} }$ | ${\href{/padicField/41.4.0.1}{4} }^{2}$ | ${\href{/padicField/43.8.0.1}{8} }$ | ${\href{/padicField/47.4.0.1}{4} }^{2}$ | ${\href{/padicField/53.8.0.1}{8} }$ | ${\href{/padicField/59.8.0.1}{8} }$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.8.31.4 | $x^{8} + 8 x^{6} + 20 x^{4} + 16 x^{2} + 18$ | $8$ | $1$ | $31$ | $C_8$ | $[3, 4, 5]$ |
\(5\) | 5.8.6.3 | $x^{8} - 50 x^{4} - 175$ | $4$ | $2$ | $6$ | $C_8$ | $[\ ]_{4}^{2}$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
* | 1.160.8t1.d.a | $1$ | $ 2^{5} \cdot 5 $ | 8.0.33554432000000.2 | $C_8$ (as 8T1) | $0$ | $-1$ |
* | 1.80.4t1.a.a | $1$ | $ 2^{4} \cdot 5 $ | 4.4.51200.1 | $C_4$ (as 4T1) | $0$ | $1$ |
* | 1.160.8t1.d.b | $1$ | $ 2^{5} \cdot 5 $ | 8.0.33554432000000.2 | $C_8$ (as 8T1) | $0$ | $-1$ |
* | 1.8.2t1.a.a | $1$ | $ 2^{3}$ | \(\Q(\sqrt{2}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
* | 1.160.8t1.d.c | $1$ | $ 2^{5} \cdot 5 $ | 8.0.33554432000000.2 | $C_8$ (as 8T1) | $0$ | $-1$ |
* | 1.80.4t1.a.b | $1$ | $ 2^{4} \cdot 5 $ | 4.4.51200.1 | $C_4$ (as 4T1) | $0$ | $1$ |
* | 1.160.8t1.d.d | $1$ | $ 2^{5} \cdot 5 $ | 8.0.33554432000000.2 | $C_8$ (as 8T1) | $0$ | $-1$ |