Normalized defining polynomial
\( x^{8} - x^{7} + 23x^{6} - 15x^{5} + 196x^{4} - 65x^{3} + 727x^{2} - 82x + 991 \)
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: | \(4241265625\) \(\medspace = 5^{6}\cdot 521^{2}\) | 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: | \(15.97\) | 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: | $5^{3/4}521^{1/2}\approx 76.32140644266818$ | ||
Ramified primes: | \(5\), \(521\) | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{20}a^{6}-\frac{1}{5}a^{5}-\frac{3}{10}a^{4}+\frac{7}{20}a^{3}+\frac{1}{20}a^{2}+\frac{1}{4}a-\frac{9}{20}$, $\frac{1}{2620}a^{7}+\frac{3}{131}a^{6}-\frac{451}{1310}a^{5}-\frac{17}{2620}a^{4}+\frac{469}{2620}a^{3}-\frac{931}{2620}a^{2}-\frac{389}{2620}a+\frac{106}{655}$
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: | \( \frac{59}{2620} a^{7} - \frac{259}{2620} a^{6} + \frac{639}{1310} a^{5} - \frac{4409}{2620} a^{4} + \frac{4469}{1310} a^{3} - \frac{6167}{655} a^{2} + \frac{9157}{1310} a - \frac{45593}{2620} \) (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: | $\frac{59}{2620}a^{7}-\frac{259}{2620}a^{6}+\frac{639}{1310}a^{5}-\frac{4409}{2620}a^{4}+\frac{4469}{1310}a^{3}-\frac{6167}{655}a^{2}+\frac{9157}{1310}a-\frac{48213}{2620}$, $\frac{77}{2620}a^{7}+\frac{821}{2620}a^{6}+\frac{381}{1310}a^{5}+\frac{2725}{524}a^{4}+\frac{214}{131}a^{3}+\frac{39547}{1310}a^{2}+\frac{3483}{655}a+\frac{148059}{2620}$, $\frac{129}{1310}a^{7}+\frac{71}{655}a^{6}+\frac{902}{655}a^{5}+\frac{557}{262}a^{4}+\frac{1725}{262}a^{3}+\frac{15093}{1310}a^{2}+\frac{14009}{1310}a+\frac{12414}{655}$ | 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: | \( 96.5540024709 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{4}\cdot 96.5540024709 \cdot 2}{10\cdot\sqrt{4241265625}}\cr\approx \mathstrut & 0.462138355047 \end{aligned}\]
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: | 8.4.46050135300625.1 |
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 | ${\href{/padicField/7.4.0.1}{4} }^{2}$ | ${\href{/padicField/11.2.0.1}{2} }^{2}{,}\,{\href{/padicField/11.1.0.1}{1} }^{4}$ | ${\href{/padicField/13.4.0.1}{4} }^{2}$ | ${\href{/padicField/17.4.0.1}{4} }^{2}$ | ${\href{/padicField/19.4.0.1}{4} }^{2}$ | ${\href{/padicField/23.4.0.1}{4} }^{2}$ | ${\href{/padicField/29.2.0.1}{2} }^{4}$ | ${\href{/padicField/31.2.0.1}{2} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }^{4}$ | ${\href{/padicField/37.4.0.1}{4} }^{2}$ | ${\href{/padicField/41.2.0.1}{2} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }^{4}$ | ${\href{/padicField/43.4.0.1}{4} }^{2}$ | ${\href{/padicField/47.4.0.1}{4} }^{2}$ | ${\href{/padicField/53.4.0.1}{4} }^{2}$ | ${\href{/padicField/59.2.0.1}{2} }^{4}$ |
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 |
---|---|---|---|---|---|---|---|
\(5\) | 5.4.3.2 | $x^{4} + 5$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ |
5.4.3.2 | $x^{4} + 5$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ | |
\(521\) | $\Q_{521}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{521}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{521}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{521}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |