Normalized defining polynomial
\( x^{8} - 4x^{7} - 144x^{6} + 446x^{5} + 6020x^{4} - 12788x^{3} - 85043x^{2} + 91512x + 171031 \)
Invariants
Degree: | $8$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[8, 0]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(2474546022446481\) \(\medspace = 3^{4}\cdot 2351^{4}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(83.98\) | 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: | $3^{1/2}2351^{1/2}\approx 83.98214095865859$ | ||
Ramified primes: | \(3\), \(2351\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
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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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2}a^{4}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{5}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{5797566}a^{6}-\frac{1}{1932522}a^{5}-\frac{401213}{2898783}a^{4}+\frac{1604857}{5797566}a^{3}+\frac{357371}{2898783}a^{2}-\frac{1517171}{5797566}a+\frac{779348}{2898783}$, $\frac{1}{5797566}a^{7}-\frac{802435}{5797566}a^{5}-\frac{802421}{5797566}a^{4}-\frac{268253}{5797566}a^{3}+\frac{627055}{5797566}a^{2}+\frac{2804749}{5797566}a-\frac{186913}{966261}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{3}$, which has order $3$
Unit group
Rank: | $7$ | 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: | $\frac{2605}{5797566}a^{6}-\frac{2605}{1932522}a^{5}-\frac{297187}{5797566}a^{4}+\frac{607399}{5797566}a^{3}+\frac{3340895}{2898783}a^{2}-\frac{3493396}{2898783}a-\frac{12387035}{5797566}$, $\frac{887}{2898783}a^{6}-\frac{887}{966261}a^{5}-\frac{201271}{5797566}a^{4}+\frac{205706}{2898783}a^{3}+\frac{2041460}{2898783}a^{2}-\frac{4289513}{5797566}a+\frac{2586505}{5797566}$, $\frac{775}{966261}a^{6}-\frac{775}{322087}a^{5}-\frac{182393}{1932522}a^{4}+\frac{186268}{966261}a^{3}+\frac{2190019}{966261}a^{2}-\frac{4567081}{1932522}a-\frac{10302571}{1932522}$, $\frac{551}{2898783}a^{7}-\frac{50}{966261}a^{6}-\frac{153655}{5797566}a^{5}-\frac{5158}{2898783}a^{4}+\frac{2799752}{2898783}a^{3}-\frac{1707185}{5797566}a^{2}-\frac{57202571}{5797566}a+\frac{4565154}{322087}$, $\frac{2045}{5797566}a^{7}+\frac{2101}{2898783}a^{6}-\frac{281003}{5797566}a^{5}-\frac{124865}{966261}a^{4}+\frac{9028807}{5797566}a^{3}+\frac{24462349}{5797566}a^{2}-\frac{25493944}{2898783}a-\frac{60069805}{5797566}$, $\frac{10747}{2898783}a^{7}-\frac{715}{21159}a^{6}-\frac{2090093}{5797566}a^{5}+\frac{3365024}{966261}a^{4}+\frac{13102664}{2898783}a^{3}-\frac{400845239}{5797566}a^{2}+\frac{201556687}{5797566}a+\frac{391684066}{2898783}$, $\frac{39961}{5797566}a^{7}-\frac{76301}{1932522}a^{6}-\frac{2439173}{2898783}a^{5}+\frac{23214895}{5797566}a^{4}+\frac{72685352}{2898783}a^{3}-\frac{445023485}{5797566}a^{2}-\frac{611031334}{2898783}a-\frac{109583903}{966261}$ | 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: | \( 54052.161116 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{8}\cdot(2\pi)^{0}\cdot 54052.161116 \cdot 3}{2\cdot\sqrt{2474546022446481}}\cr\approx \mathstrut & 0.41725016712 \end{aligned}\]
Galois group
A solvable group of order 24 |
The 5 conjugacy class representatives for $S_4$ |
Character table for $S_4$ |
Intermediate fields
\(\Q(\sqrt{7053}) \), 4.4.7053.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Galois closure: | deg 24 |
Degree 4 sibling: | 4.4.7053.1 |
Degree 6 siblings: | 6.6.49744809.1, 6.6.350850137877.2 |
Degree 12 siblings: | deg 12, deg 12 |
Minimal sibling: | 4.4.7053.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | ${\href{/padicField/2.4.0.1}{4} }^{2}$ | R | ${\href{/padicField/5.4.0.1}{4} }^{2}$ | ${\href{/padicField/7.3.0.1}{3} }^{2}{,}\,{\href{/padicField/7.1.0.1}{1} }^{2}$ | ${\href{/padicField/11.4.0.1}{4} }^{2}$ | ${\href{/padicField/13.2.0.1}{2} }^{4}$ | ${\href{/padicField/17.3.0.1}{3} }^{2}{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ | ${\href{/padicField/19.3.0.1}{3} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | ${\href{/padicField/23.4.0.1}{4} }^{2}$ | ${\href{/padicField/29.3.0.1}{3} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.3.0.1}{3} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.4.0.1}{4} }^{2}$ | ${\href{/padicField/41.2.0.1}{2} }^{4}$ | ${\href{/padicField/43.2.0.1}{2} }^{4}$ | ${\href{/padicField/47.3.0.1}{3} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | ${\href{/padicField/53.3.0.1}{3} }^{2}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | ${\href{/padicField/59.4.0.1}{4} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(3\) | 3.4.2.1 | $x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
3.4.2.1 | $x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
\(2351\) | Deg $4$ | $2$ | $2$ | $2$ | |||
Deg $4$ | $2$ | $2$ | $2$ |