Normalized defining polynomial
\( x^{8} - 3x^{7} + 7x^{6} - 22x^{5} + 36x^{4} - 49x^{3} + 76x^{2} - 60x + 25 \)
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: | \(20671250625\) \(\medspace = 3^{8}\cdot 5^{4}\cdot 71^{2}\) | 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: | \(19.47\) | 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: | $3^{4/3}5^{1/2}71^{1/2}\approx 81.52219216048373$ | ||
Ramified primes: | \(3\), \(5\), \(71\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q\) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{3}a^{5}+\frac{1}{3}a^{3}+\frac{1}{3}a^{2}+\frac{1}{3}$, $\frac{1}{3}a^{6}+\frac{1}{3}a^{4}+\frac{1}{3}a^{3}+\frac{1}{3}a$, $\frac{1}{15}a^{7}+\frac{2}{15}a^{6}+\frac{2}{15}a^{5}+\frac{1}{5}a^{4}+\frac{2}{5}a^{3}-\frac{4}{15}a^{2}-\frac{4}{15}a-\frac{1}{3}$
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: | \( -1 \) (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)
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Fundamental units: | $\frac{1}{15}a^{7}-\frac{1}{5}a^{6}+\frac{7}{15}a^{5}-\frac{17}{15}a^{4}+\frac{7}{5}a^{3}-\frac{29}{15}a^{2}+\frac{7}{5}a$, $\frac{1}{3}a^{6}-\frac{4}{3}a^{5}+\frac{7}{3}a^{4}-5a^{3}+\frac{26}{3}a^{2}-\frac{20}{3}a+\frac{8}{3}$, $\frac{2}{15}a^{7}-\frac{1}{15}a^{6}+\frac{14}{15}a^{5}-\frac{44}{15}a^{4}+\frac{32}{15}a^{3}-\frac{133}{15}a^{2}+\frac{137}{15}a-5$ | sage: UK.fundamental_units()
gp: K.fu
magma: [K|fUK(g): g in Generators(UK)];
oscar: [K(fUK(a)) for a in gens(UK)]
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Regulator: | \( 50.740786861 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{4}\cdot 50.740786861 \cdot 2}{2\cdot\sqrt{20671250625}}\cr\approx \mathstrut & 0.55003876073 \end{aligned}\]
Galois group
$C_2^3:A_4$ (as 8T33):
A solvable group of order 96 |
The 10 conjugacy class representatives for $C_2^4:C_6$ |
Character table for $C_2^4:C_6$ |
Intermediate fields
\(\Q(\sqrt{5}) \) |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 8 sibling: | data not computed |
Degree 12 siblings: | data not computed |
Degree 16 sibling: | data not computed |
Degree 24 siblings: | data not computed |
Degree 32 sibling: | 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 | ${\href{/padicField/2.6.0.1}{6} }{,}\,{\href{/padicField/2.2.0.1}{2} }$ | R | R | ${\href{/padicField/7.6.0.1}{6} }{,}\,{\href{/padicField/7.2.0.1}{2} }$ | ${\href{/padicField/11.3.0.1}{3} }^{2}{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ | ${\href{/padicField/13.6.0.1}{6} }{,}\,{\href{/padicField/13.2.0.1}{2} }$ | ${\href{/padicField/17.4.0.1}{4} }^{2}$ | ${\href{/padicField/19.2.0.1}{2} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }^{4}$ | ${\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.2.0.1}{2} }$ | ${\href{/padicField/29.3.0.1}{3} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.3.0.1}{3} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.4.0.1}{4} }^{2}$ | ${\href{/padicField/41.3.0.1}{3} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.2.0.1}{2} }$ | ${\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.2.0.1}{2} }$ | ${\href{/padicField/53.4.0.1}{4} }^{2}$ | ${\href{/padicField/59.3.0.1}{3} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(3\) | 3.2.0.1 | $x^{2} + 2 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
3.6.8.3 | $x^{6} + 18 x^{5} + 114 x^{4} + 326 x^{3} + 570 x^{2} + 528 x + 197$ | $3$ | $2$ | $8$ | $C_6$ | $[2]^{2}$ | |
\(5\) | 5.2.1.1 | $x^{2} + 5$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
5.6.3.1 | $x^{6} + 60 x^{5} + 1221 x^{4} + 8846 x^{3} + 9864 x^{2} + 29208 x + 29309$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
\(71\) | $\Q_{71}$ | $x + 64$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{71}$ | $x + 64$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{71}$ | $x + 64$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{71}$ | $x + 64$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
71.4.2.1 | $x^{4} + 138 x^{3} + 4917 x^{2} + 10764 x + 342127$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |