Normalized defining polynomial
\( x^{8} - 4x^{7} + 7x^{6} - 7x^{5} - 7x^{4} + 21x^{3} - 2x^{2} - 9x + 3 \)
Invariants
Degree: | $8$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[4, 2]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(5082629877\) \(\medspace = 3^{4}\cdot 13^{7}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(16.34\) | 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: | $3^{1/2}13^{7/8}\approx 16.340338938923477$ | ||
Ramified primes: | \(3\), \(13\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{13}) \) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
This field is not Galois over $\Q$. | |||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$
Monogenic: | Yes | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$
Unit group
Rank: | $5$ | 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: | $a^{6}-2a^{5}+3a^{4}-2a^{3}-8a^{2}+a+2$, $a^{6}-4a^{5}+8a^{4}-10a^{3}-a^{2}+13a-5$, $3a^{7}-10a^{6}+14a^{5}-10a^{4}-31a^{3}+47a^{2}+24a-17$, $2a^{7}-7a^{6}+11a^{5}-10a^{4}-16a^{3}+30a^{2}+10a-10$, $a^{6}-3a^{5}+6a^{4}-7a^{3}-3a^{2}+6a-5$ | 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: | \( 123.222850641 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{2}\cdot 123.222850641 \cdot 1}{2\cdot\sqrt{5082629877}}\cr\approx \mathstrut & 0.545879434219 \end{aligned}\]
Galois group
$\OD_{16}:C_2$ (as 8T16):
A solvable group of order 32 |
The 11 conjugacy class representatives for $(C_8:C_2):C_2$ |
Character table for $(C_8:C_2):C_2$ |
Intermediate fields
\(\Q(\sqrt{13}) \), 4.4.19773.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Galois closure: | data not computed |
Degree 8 sibling: | data not computed |
Degree 16 siblings: | 16.4.25833126466573035129.1, 16.0.2870347385174781681.1, 16.0.25833126466573035129.1 |
Minimal sibling: | 8.0.564736653.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | ${\href{/padicField/2.8.0.1}{8} }$ | R | ${\href{/padicField/5.8.0.1}{8} }$ | ${\href{/padicField/7.8.0.1}{8} }$ | ${\href{/padicField/11.8.0.1}{8} }$ | R | ${\href{/padicField/17.2.0.1}{2} }^{2}{,}\,{\href{/padicField/17.1.0.1}{1} }^{4}$ | ${\href{/padicField/19.8.0.1}{8} }$ | ${\href{/padicField/23.2.0.1}{2} }^{2}{,}\,{\href{/padicField/23.1.0.1}{1} }^{4}$ | ${\href{/padicField/29.2.0.1}{2} }^{4}$ | ${\href{/padicField/31.8.0.1}{8} }$ | ${\href{/padicField/37.8.0.1}{8} }$ | ${\href{/padicField/41.8.0.1}{8} }$ | ${\href{/padicField/43.4.0.1}{4} }^{2}$ | ${\href{/padicField/47.8.0.1}{8} }$ | ${\href{/padicField/53.2.0.1}{2} }^{4}$ | ${\href{/padicField/59.8.0.1}{8} }$ |
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 |
---|---|---|---|---|---|---|---|
\(3\) | 3.2.1.2 | $x^{2} + 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
3.2.1.2 | $x^{2} + 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
3.4.2.1 | $x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
\(13\) | 13.8.7.2 | $x^{8} + 13$ | $8$ | $1$ | $7$ | $C_8:C_2$ | $[\ ]_{8}^{2}$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
1.3.2t1.a.a | $1$ | $ 3 $ | \(\Q(\sqrt{-3}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
* | 1.13.2t1.a.a | $1$ | $ 13 $ | \(\Q(\sqrt{13}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
1.39.2t1.a.a | $1$ | $ 3 \cdot 13 $ | \(\Q(\sqrt{-39}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
1.13.4t1.a.a | $1$ | $ 13 $ | 4.0.2197.1 | $C_4$ (as 4T1) | $0$ | $-1$ | |
* | 1.39.4t1.a.a | $1$ | $ 3 \cdot 13 $ | 4.4.19773.1 | $C_4$ (as 4T1) | $0$ | $1$ |
1.13.4t1.a.b | $1$ | $ 13 $ | 4.0.2197.1 | $C_4$ (as 4T1) | $0$ | $-1$ | |
* | 1.39.4t1.a.b | $1$ | $ 3 \cdot 13 $ | 4.4.19773.1 | $C_4$ (as 4T1) | $0$ | $1$ |
2.507.4t3.c.a | $2$ | $ 3 \cdot 13^{2}$ | 4.0.19773.1 | $D_{4}$ (as 4T3) | $1$ | $0$ | |
2.39.4t3.a.a | $2$ | $ 3 \cdot 13 $ | 4.0.117.1 | $D_{4}$ (as 4T3) | $1$ | $0$ | |
* | 4.257049.8t16.a.a | $4$ | $ 3^{2} \cdot 13^{4}$ | 8.4.5082629877.1 | $(C_8:C_2):C_2$ (as 8T16) | $1$ | $0$ |