Normalized defining polynomial
\( x^{8} - 2x^{7} + 12x^{6} - 20x^{5} + 111x^{4} - 10x^{3} + 543x^{2} + 139x + 881 \)
Invariants
Degree: | $8$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[0, 4]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(3178140625\) \(\medspace = 5^{6}\cdot 11^{2}\cdot 41^{2}\) | 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: | \(15.41\) | 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: | $5^{3/4}11^{1/2}41^{1/2}\approx 71.0093887402365$ | ||
Ramified primes: | \(5\), \(11\), \(41\) | 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)
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This field is not Galois over $\Q$. | |||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{11}a^{6}-\frac{5}{11}a^{5}+\frac{5}{11}a^{3}-\frac{3}{11}a^{2}-\frac{4}{11}a-\frac{2}{11}$, $\frac{1}{383581}a^{7}+\frac{741}{34871}a^{6}-\frac{43882}{383581}a^{5}+\frac{41365}{383581}a^{4}+\frac{7387}{34871}a^{3}+\frac{113666}{383581}a^{2}+\frac{2056}{34871}a+\frac{60281}{383581}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{2}$, which has order $2$
Unit group
Rank: | $3$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
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Torsion generator: | \( -\frac{2834}{383581} a^{7} + \frac{19539}{383581} a^{6} - \frac{58140}{383581} a^{5} + \frac{147376}{383581} a^{4} - \frac{377835}{383581} a^{3} + \frac{915500}{383581} a^{2} - \frac{454169}{383581} a + \frac{2333032}{383581} \) (order $10$) | sage: UK.torsion_generator()
gp: K.tu[2]
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
oscar: torsion_units_generator(OK)
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Fundamental units: | $\frac{1588}{383581}a^{7}+\frac{6647}{383581}a^{6}-\frac{12358}{383581}a^{5}+\frac{95269}{383581}a^{4}-\frac{91197}{383581}a^{3}+\frac{1055442}{383581}a^{2}+\frac{206304}{383581}a+\frac{3457562}{383581}$, $\frac{12894}{383581}a^{7}-\frac{37071}{383581}a^{6}+\frac{141822}{383581}a^{5}-\frac{200861}{383581}a^{4}+\frac{1144435}{383581}a^{3}+\frac{51016}{383581}a^{2}+\frac{2913695}{383581}a+\frac{581431}{383581}$, $\frac{27541}{383581}a^{7}+\frac{161548}{383581}a^{6}-\frac{239141}{383581}a^{5}+\frac{1532219}{383581}a^{4}-\frac{714550}{383581}a^{3}+\frac{18657008}{383581}a^{2}+\frac{12310917}{383581}a+\frac{52087985}{383581}$ | 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: | \( 82.6261050011 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{4}\cdot 82.6261050011 \cdot 2}{10\cdot\sqrt{3178140625}}\cr\approx \mathstrut & 0.456856906577 \end{aligned}\]
Galois group
$C_2^3:C_4$ (as 8T20):
A solvable group of order 32 |
The 11 conjugacy class representatives for $C_2^3: C_4$ |
Character table for $C_2^3: C_4$ |
Intermediate fields
\(\Q(\sqrt{5}) \), \(\Q(\zeta_{5})\) |
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: | 8.4.25857479250625.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | ${\href{/padicField/2.4.0.1}{4} }^{2}$ | ${\href{/padicField/3.4.0.1}{4} }^{2}$ | R | ${\href{/padicField/7.4.0.1}{4} }^{2}$ | R | ${\href{/padicField/13.4.0.1}{4} }^{2}$ | ${\href{/padicField/17.4.0.1}{4} }^{2}$ | ${\href{/padicField/19.4.0.1}{4} }^{2}$ | ${\href{/padicField/23.4.0.1}{4} }^{2}$ | ${\href{/padicField/29.4.0.1}{4} }^{2}$ | ${\href{/padicField/31.2.0.1}{2} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }^{4}$ | ${\href{/padicField/37.4.0.1}{4} }^{2}$ | R | ${\href{/padicField/43.4.0.1}{4} }^{2}$ | ${\href{/padicField/47.4.0.1}{4} }^{2}$ | ${\href{/padicField/53.4.0.1}{4} }^{2}$ | ${\href{/padicField/59.2.0.1}{2} }^{4}$ |
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 |
---|---|---|---|---|---|---|---|
\(5\) | 5.4.3.2 | $x^{4} + 5$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ |
5.4.3.2 | $x^{4} + 5$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ | |
\(11\) | $\Q_{11}$ | $x + 9$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{11}$ | $x + 9$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
11.2.1.1 | $x^{2} + 22$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
11.2.1.2 | $x^{2} + 11$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
11.2.0.1 | $x^{2} + 7 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
\(41\) | $\Q_{41}$ | $x + 35$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{41}$ | $x + 35$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{41}$ | $x + 35$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{41}$ | $x + 35$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
41.2.1.1 | $x^{2} + 41$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
41.2.1.1 | $x^{2} + 41$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |