Normalized defining polynomial
\( x^{8} - 2x^{7} + 28x^{6} - 120x^{5} + 491x^{4} - 1870x^{3} + 5673x^{2} - 6643x + 8771 \)
Invariants
Degree: | $8$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[0, 4]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(6472063200625\) \(\medspace = 5^{4}\cdot 11^{4}\cdot 29^{4}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(39.94\) | 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: | $5^{1/2}11^{1/2}29^{1/2}\approx 39.93745109543172$ | ||
Ramified primes: | \(5\), \(11\), \(29\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q\) | ||
$\card{ \Gal(K/\Q) }$: | $8$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
This field is Galois over $\Q$. | |||
This is a CM field. | |||
Reflex fields: | 4.0.87725.1$^{4}$, 4.0.508805.3$^{4}$ |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2755}a^{6}+\frac{1057}{2755}a^{5}-\frac{17}{95}a^{4}-\frac{18}{551}a^{3}-\frac{337}{2755}a^{2}+\frac{37}{2755}a-\frac{256}{2755}$, $\frac{1}{263799515}a^{7}+\frac{39702}{263799515}a^{6}-\frac{3427379}{37685645}a^{5}-\frac{8034932}{52759903}a^{4}+\frac{1806472}{13884185}a^{3}-\frac{34920068}{263799515}a^{2}-\frac{98328941}{263799515}a+\frac{1708496}{7537129}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{4}\times C_{4}$, which has order $16$
Unit group
Rank: | $3$ | 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: | $\frac{2916}{13884185}a^{7}+\frac{39}{95753}a^{6}+\frac{9569}{1983455}a^{5}-\frac{109204}{13884185}a^{4}+\frac{430203}{13884185}a^{3}-\frac{1933299}{13884185}a^{2}+\frac{401339}{2776837}a+\frac{2656036}{1983455}$, $\frac{1002}{7537129}a^{7}+\frac{2872}{7537129}a^{6}+\frac{2442}{7537129}a^{5}+\frac{173}{396691}a^{4}-\frac{374086}{7537129}a^{3}+\frac{1794604}{7537129}a^{2}-\frac{2314123}{7537129}a+\frac{42644857}{7537129}$, $\frac{84898}{263799515}a^{7}-\frac{15342}{52759903}a^{6}+\frac{301067}{37685645}a^{5}-\frac{3826972}{263799515}a^{4}+\frac{15191864}{263799515}a^{3}-\frac{60770362}{263799515}a^{2}+\frac{13319366}{52759903}a-\frac{19307332}{37685645}$ | 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: | \( 40.3281811419 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{4}\cdot 40.3281811419 \cdot 16}{2\cdot\sqrt{6472063200625}}\cr\approx \mathstrut & 0.197649955451 \end{aligned}\]
Galois group
A solvable group of order 8 |
The 5 conjugacy class representatives for $D_4$ |
Character table for $D_4$ |
Intermediate fields
\(\Q(\sqrt{145}) \), \(\Q(\sqrt{29}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{5}, \sqrt{29})\), 4.0.508805.3 x2, 4.0.87725.1 x2 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 4 siblings: | 4.0.87725.1, 4.0.508805.3 |
Minimal sibling: | 4.0.87725.1 |
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 | ${\href{/padicField/7.2.0.1}{2} }^{4}$ | R | ${\href{/padicField/13.2.0.1}{2} }^{4}$ | ${\href{/padicField/17.4.0.1}{4} }^{2}$ | ${\href{/padicField/19.2.0.1}{2} }^{4}$ | ${\href{/padicField/23.2.0.1}{2} }^{4}$ | R | ${\href{/padicField/31.2.0.1}{2} }^{4}$ | ${\href{/padicField/37.4.0.1}{4} }^{2}$ | ${\href{/padicField/41.2.0.1}{2} }^{4}$ | ${\href{/padicField/43.4.0.1}{4} }^{2}$ | ${\href{/padicField/47.4.0.1}{4} }^{2}$ | ${\href{/padicField/53.2.0.1}{2} }^{4}$ | ${\href{/padicField/59.2.0.1}{2} }^{4}$ |
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 |
---|---|---|---|---|---|---|---|
\(5\) | 5.4.2.1 | $x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
5.4.2.1 | $x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
\(11\) | 11.4.2.1 | $x^{4} + 14 x^{3} + 75 x^{2} + 182 x + 620$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
11.4.2.1 | $x^{4} + 14 x^{3} + 75 x^{2} + 182 x + 620$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
\(29\) | 29.2.1.1 | $x^{2} + 29$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
29.2.1.1 | $x^{2} + 29$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
29.2.1.1 | $x^{2} + 29$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
29.2.1.1 | $x^{2} + 29$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |