Normalized defining polynomial
\( x^{8} - 2x^{7} - 5x^{6} + 10x^{5} + 15x^{4} - 46x^{3} - 17x^{2} + 68x + 73 \)
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: | \(424030464\) \(\medspace = 2^{8}\cdot 3^{4}\cdot 11^{2}\cdot 13^{2}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(11.98\) | 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: | $2\cdot 3^{1/2}11^{1/2}13^{1/2}\approx 41.42463035441596$ | ||
Ramified primes: | \(2\), \(3\), \(11\), \(13\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q\) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{35}a^{6}+\frac{8}{35}a^{5}+\frac{1}{5}a^{4}-\frac{9}{35}a^{3}+\frac{9}{35}a^{2}-\frac{9}{35}a+\frac{16}{35}$, $\frac{1}{1295}a^{7}+\frac{2}{259}a^{6}-\frac{292}{1295}a^{5}-\frac{55}{259}a^{4}+\frac{341}{1295}a^{3}-\frac{61}{1295}a^{2}-\frac{527}{1295}a-\frac{3}{1295}$
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)
| |
Torsion generator: | \( \frac{58}{1295} a^{7} + \frac{62}{1295} a^{6} - \frac{72}{259} a^{5} - \frac{151}{1295} a^{4} + \frac{226}{259} a^{3} - \frac{86}{259} a^{2} - \frac{3889}{1295} a - \frac{1987}{1295} \) (order $12$) | 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: | $\frac{58}{1295}a^{7}+\frac{62}{1295}a^{6}-\frac{72}{259}a^{5}-\frac{151}{1295}a^{4}+\frac{226}{259}a^{3}-\frac{86}{259}a^{2}-\frac{3889}{1295}a-\frac{692}{1295}$, $\frac{261}{1295}a^{7}-\frac{276}{1295}a^{6}-\frac{176}{259}a^{5}+\frac{1263}{1295}a^{4}+\frac{66}{37}a^{3}-\frac{198}{37}a^{2}-\frac{399}{185}a-\frac{2929}{1295}$, $\frac{984}{1295}a^{7}+\frac{183}{1295}a^{6}-\frac{4574}{1295}a^{5}-\frac{204}{1295}a^{4}+\frac{2076}{185}a^{3}-\frac{1936}{185}a^{2}-\frac{1344}{37}a-\frac{34439}{1295}$ | 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: | \( 55.4013718228 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{4}\cdot 55.4013718228 \cdot 1}{12\cdot\sqrt{424030464}}\cr\approx \mathstrut & 0.349430022746 \end{aligned}\]
Galois group
$D_4:C_2^2$ (as 8T22):
A solvable group of order 32 |
The 17 conjugacy class representatives for $Q_8:C_2^2$ |
Character table for $Q_8:C_2^2$ |
Intermediate fields
\(\Q(\sqrt{-3}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{-1}) \), \(\Q(\zeta_{12})\) |
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 siblings: | data not computed |
Degree 16 siblings: | data not computed |
Minimal sibling: | This field is its own minimal sibling |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | R | ${\href{/padicField/5.2.0.1}{2} }^{4}$ | ${\href{/padicField/7.2.0.1}{2} }^{4}$ | R | R | ${\href{/padicField/17.4.0.1}{4} }^{2}$ | ${\href{/padicField/19.2.0.1}{2} }^{4}$ | ${\href{/padicField/23.2.0.1}{2} }^{4}$ | ${\href{/padicField/29.4.0.1}{4} }^{2}$ | ${\href{/padicField/31.4.0.1}{4} }^{2}$ | ${\href{/padicField/37.2.0.1}{2} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }^{4}$ | ${\href{/padicField/41.4.0.1}{4} }^{2}$ | ${\href{/padicField/43.4.0.1}{4} }^{2}$ | ${\href{/padicField/47.4.0.1}{4} }^{2}$ | ${\href{/padicField/53.2.0.1}{2} }^{4}$ | ${\href{/padicField/59.4.0.1}{4} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.8.8.1 | $x^{8} + 8 x^{7} + 32 x^{6} + 82 x^{5} + 148 x^{4} + 184 x^{3} + 137 x^{2} + 44 x + 5$ | $2$ | $4$ | $8$ | $C_4\times C_2$ | $[2]^{4}$ |
\(3\) | 3.4.2.1 | $x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
3.4.2.1 | $x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
\(11\) | 11.4.2.2 | $x^{4} - 77 x^{2} + 242$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ |
11.4.0.1 | $x^{4} + 8 x^{2} + 10 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
\(13\) | $\Q_{13}$ | $x + 11$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{13}$ | $x + 11$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
13.2.1.2 | $x^{2} + 26$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
13.2.1.1 | $x^{2} + 13$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
13.2.0.1 | $x^{2} + 12 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
1.13.2t1.a.a | $1$ | $ 13 $ | \(\Q(\sqrt{13}) \) | $C_2$ (as 2T1) | $1$ | $1$ | |
1.429.2t1.a.a | $1$ | $ 3 \cdot 11 \cdot 13 $ | \(\Q(\sqrt{429}) \) | $C_2$ (as 2T1) | $1$ | $1$ | |
1.33.2t1.a.a | $1$ | $ 3 \cdot 11 $ | \(\Q(\sqrt{33}) \) | $C_2$ (as 2T1) | $1$ | $1$ | |
1.11.2t1.a.a | $1$ | $ 11 $ | \(\Q(\sqrt{-11}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
1.143.2t1.a.a | $1$ | $ 11 \cdot 13 $ | \(\Q(\sqrt{-143}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
* | 1.3.2t1.a.a | $1$ | $ 3 $ | \(\Q(\sqrt{-3}) \) | $C_2$ (as 2T1) | $1$ | $-1$ |
1.39.2t1.a.a | $1$ | $ 3 \cdot 13 $ | \(\Q(\sqrt{-39}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
1.1716.2t1.a.a | $1$ | $ 2^{2} \cdot 3 \cdot 11 \cdot 13 $ | \(\Q(\sqrt{-429}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
1.132.2t1.a.a | $1$ | $ 2^{2} \cdot 3 \cdot 11 $ | \(\Q(\sqrt{-33}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
1.52.2t1.a.a | $1$ | $ 2^{2} \cdot 13 $ | \(\Q(\sqrt{-13}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
* | 1.4.2t1.a.a | $1$ | $ 2^{2}$ | \(\Q(\sqrt{-1}) \) | $C_2$ (as 2T1) | $1$ | $-1$ |
* | 1.12.2t1.a.a | $1$ | $ 2^{2} \cdot 3 $ | \(\Q(\sqrt{3}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
1.156.2t1.a.a | $1$ | $ 2^{2} \cdot 3 \cdot 13 $ | \(\Q(\sqrt{39}) \) | $C_2$ (as 2T1) | $1$ | $1$ | |
1.44.2t1.a.a | $1$ | $ 2^{2} \cdot 11 $ | \(\Q(\sqrt{11}) \) | $C_2$ (as 2T1) | $1$ | $1$ | |
1.572.2t1.a.a | $1$ | $ 2^{2} \cdot 11 \cdot 13 $ | \(\Q(\sqrt{143}) \) | $C_2$ (as 2T1) | $1$ | $1$ | |
* | 4.2944656.8t22.f.a | $4$ | $ 2^{4} \cdot 3^{2} \cdot 11^{2} \cdot 13^{2}$ | 8.0.424030464.1 | $Q_8:C_2^2$ (as 8T22) | $1$ | $0$ |