Normalized defining polynomial
\( x^{8} - x^{7} - 9x^{6} + 4x^{5} + 41x^{4} + 10x^{3} - 210x^{2} + 260x - 95 \)
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)
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Discriminant: | \(18530015625\) \(\medspace = 3^{4}\cdot 5^{6}\cdot 11^{4}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
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Root discriminant: | \(19.21\) | 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))
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Galois root discriminant: | $3^{1/2}5^{3/4}11^{1/2}\approx 19.208102881010017$ | ||
Ramified primes: | \(3\), \(5\), \(11\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q\) | ||
$\card{ \Aut(K/\Q) }$: | $4$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{11}a^{6}-\frac{1}{11}a^{4}+\frac{3}{11}a^{3}+\frac{3}{11}a^{2}+\frac{4}{11}a+\frac{5}{11}$, $\frac{1}{11}a^{7}-\frac{1}{11}a^{5}+\frac{3}{11}a^{4}+\frac{3}{11}a^{3}+\frac{4}{11}a^{2}+\frac{5}{11}a$
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)
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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: | $\frac{8}{11}a^{7}+\frac{6}{11}a^{6}-\frac{63}{11}a^{5}-\frac{81}{11}a^{4}+\frac{196}{11}a^{3}+\frac{446}{11}a^{2}-\frac{915}{11}a+\frac{382}{11}$, $\frac{10}{11}a^{7}+\frac{8}{11}a^{6}-\frac{76}{11}a^{5}-9a^{4}+\frac{230}{11}a^{3}+\frac{526}{11}a^{2}-\frac{1128}{11}a+\frac{557}{11}$, $\frac{3}{11}a^{7}-\frac{3}{11}a^{6}-\frac{36}{11}a^{5}-\frac{10}{11}a^{4}+14a^{3}+\frac{168}{11}a^{2}-\frac{613}{11}a+\frac{337}{11}$, $\frac{38}{11}a^{7}+\frac{24}{11}a^{6}-\frac{302}{11}a^{5}-\frac{339}{11}a^{4}+\frac{1000}{11}a^{3}+\frac{1995}{11}a^{2}-429a+\frac{2243}{11}$, $\frac{24}{11}a^{7}+\frac{15}{11}a^{6}-\frac{189}{11}a^{5}-\frac{207}{11}a^{4}+\frac{634}{11}a^{3}+\frac{1230}{11}a^{2}-\frac{3021}{11}a+\frac{1461}{11}$ | sage: UK.fundamental_units()
gp: K.fu
magma: [K|fUK(g): g in Generators(UK)];
oscar: [K(fUK(a)) for a in gens(UK)]
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Regulator: | \( 98.701164995 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{2}\cdot 98.701164995 \cdot 2}{2\cdot\sqrt{18530015625}}\cr\approx \mathstrut & 0.45799855247 \end{aligned}\]
Galois group
$C_2^2:C_4$ (as 8T10):
A solvable group of order 16 |
The 10 conjugacy class representatives for $C_2^2:C_4$ |
Character table for $C_2^2:C_4$ |
Intermediate fields
\(\Q(\sqrt{5}) \), 4.4.15125.1, 4.2.2475.1, 4.2.12375.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Galois closure: | 16.0.343361479062744140625.5 |
Degree 8 sibling: | 8.0.153140625.2 |
Minimal sibling: | 8.0.153140625.2 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | ${\href{/padicField/2.4.0.1}{4} }^{2}$ | R | R | ${\href{/padicField/7.4.0.1}{4} }^{2}$ | R | ${\href{/padicField/13.4.0.1}{4} }^{2}$ | ${\href{/padicField/17.4.0.1}{4} }^{2}$ | ${\href{/padicField/19.2.0.1}{2} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }^{4}$ | ${\href{/padicField/23.4.0.1}{4} }^{2}$ | ${\href{/padicField/29.2.0.1}{2} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }^{4}$ | ${\href{/padicField/31.2.0.1}{2} }^{4}$ | ${\href{/padicField/37.4.0.1}{4} }^{2}$ | ${\href{/padicField/41.2.0.1}{2} }^{4}$ | ${\href{/padicField/43.4.0.1}{4} }^{2}$ | ${\href{/padicField/47.4.0.1}{4} }^{2}$ | ${\href{/padicField/53.4.0.1}{4} }^{2}$ | ${\href{/padicField/59.2.0.1}{2} }^{4}$ |
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 |
---|---|---|---|---|---|---|---|
\(3\) | 3.8.4.1 | $x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |
\(5\) | 5.4.3.2 | $x^{4} + 5$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ |
5.4.3.2 | $x^{4} + 5$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ | |
\(11\) | 11.2.1.2 | $x^{2} + 11$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
11.2.1.2 | $x^{2} + 11$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
11.2.1.2 | $x^{2} + 11$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
11.2.1.2 | $x^{2} + 11$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |