Normalized defining polynomial
\( x^{8} + 8x^{6} - 92x^{4} + 112x^{2} + 98 \)
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: | \(105226698752\) \(\medspace = 2^{31}\cdot 7^{2}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(23.87\) | 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: | $2^{31/8}7^{1/2}\approx 38.81863439296463$ | ||
Ramified primes: | \(2\), \(7\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{2}) \) | ||
$\card{ \Aut(K/\Q) }$: | $4$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{4711}a^{6}-\frac{76}{4711}a^{4}+\frac{1581}{4711}a^{2}-\frac{112}{673}$, $\frac{1}{4711}a^{7}-\frac{76}{4711}a^{5}+\frac{1581}{4711}a^{3}-\frac{112}{673}a$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{2}$, which has order $2$
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: | $\frac{8}{673}a^{6}+\frac{65}{673}a^{4}-\frac{812}{673}a^{2}+\frac{1131}{673}$, $\frac{109}{4711}a^{6}+\frac{1138}{4711}a^{4}-\frac{6689}{4711}a^{2}-\frac{767}{673}$, $\frac{2}{4711}a^{6}-\frac{152}{4711}a^{4}-\frac{1549}{4711}a^{2}+\frac{1795}{673}$, $\frac{107}{4711}a^{7}+\frac{216}{4711}a^{6}+\frac{1290}{4711}a^{5}+\frac{2428}{4711}a^{4}-\frac{5140}{4711}a^{3}-\frac{11829}{4711}a^{2}-\frac{1889}{673}a-\frac{1983}{673}$, $\frac{8}{673}a^{7}-\frac{216}{4711}a^{6}+\frac{65}{673}a^{5}-\frac{2428}{4711}a^{4}-\frac{812}{673}a^{3}+\frac{11829}{4711}a^{2}-\frac{215}{673}a+\frac{637}{673}$ | 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: | \( 234.278040356 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{2}\cdot 234.278040356 \cdot 2}{2\cdot\sqrt{105226698752}}\cr\approx \mathstrut & 0.456192697713 \end{aligned}\]
Galois group
$\OD_{16}$ (as 8T7):
A solvable group of order 16 |
The 10 conjugacy class representatives for $C_8:C_2$ |
Character table for $C_8:C_2$ |
Intermediate fields
\(\Q(\sqrt{2}) \), \(\Q(\zeta_{16})^+\) |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Galois closure: | deg 16 |
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 | ${\href{/padicField/3.8.0.1}{8} }$ | ${\href{/padicField/5.8.0.1}{8} }$ | R | ${\href{/padicField/11.8.0.1}{8} }$ | ${\href{/padicField/13.8.0.1}{8} }$ | ${\href{/padicField/17.2.0.1}{2} }^{2}{,}\,{\href{/padicField/17.1.0.1}{1} }^{4}$ | ${\href{/padicField/19.8.0.1}{8} }$ | ${\href{/padicField/23.4.0.1}{4} }^{2}$ | ${\href{/padicField/29.8.0.1}{8} }$ | ${\href{/padicField/31.2.0.1}{2} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }^{4}$ | ${\href{/padicField/37.8.0.1}{8} }$ | ${\href{/padicField/41.4.0.1}{4} }^{2}$ | ${\href{/padicField/43.8.0.1}{8} }$ | ${\href{/padicField/47.2.0.1}{2} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }^{4}$ | ${\href{/padicField/53.8.0.1}{8} }$ | ${\href{/padicField/59.8.0.1}{8} }$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.8.31.5 | $x^{8} + 8 x^{6} + 16 x^{5} + 4 x^{4} + 34$ | $8$ | $1$ | $31$ | $C_8$ | $[3, 4, 5]$ |
\(7\) | 7.4.2.2 | $x^{4} - 42 x^{2} + 147$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ |
7.4.0.1 | $x^{4} + 5 x^{2} + 4 x + 3$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
1.7.2t1.a.a | $1$ | $ 7 $ | \(\Q(\sqrt{-7}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
1.56.2t1.b.a | $1$ | $ 2^{3} \cdot 7 $ | \(\Q(\sqrt{-14}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
* | 1.8.2t1.a.a | $1$ | $ 2^{3}$ | \(\Q(\sqrt{2}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
* | 1.16.4t1.a.a | $1$ | $ 2^{4}$ | \(\Q(\zeta_{16})^+\) | $C_4$ (as 4T1) | $0$ | $1$ |
1.112.4t1.b.a | $1$ | $ 2^{4} \cdot 7 $ | 4.0.100352.5 | $C_4$ (as 4T1) | $0$ | $-1$ | |
1.112.4t1.b.b | $1$ | $ 2^{4} \cdot 7 $ | 4.0.100352.5 | $C_4$ (as 4T1) | $0$ | $-1$ | |
* | 1.16.4t1.a.b | $1$ | $ 2^{4}$ | \(\Q(\zeta_{16})^+\) | $C_4$ (as 4T1) | $0$ | $1$ |
* | 2.7168.8t7.c.a | $2$ | $ 2^{10} \cdot 7 $ | 8.4.105226698752.6 | $C_8:C_2$ (as 8T7) | $0$ | $0$ |
* | 2.7168.8t7.c.b | $2$ | $ 2^{10} \cdot 7 $ | 8.4.105226698752.6 | $C_8:C_2$ (as 8T7) | $0$ | $0$ |