Normalized defining polynomial
\( x^{8} - 2x^{7} + 10x^{6} + 38x^{5} - 526x^{4} + 472x^{3} + 335x^{2} - 1108x + 551 \)
Invariants
Degree: | $8$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[4, 2]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(20794740015625\) \(\medspace = 5^{6}\cdot 191^{4}\) | 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: | \(46.21\) | 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: | $5^{3/4}191^{1/2}\approx 46.210874461670805$ | ||
Ramified primes: | \(5\), \(191\) | 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: | \(\Q\) | ||
$\card{ \Aut(K/\Q) }$: | $4$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2}a^{4}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{5}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}$, $\frac{1}{2390714842}a^{7}-\frac{271525403}{2390714842}a^{6}-\frac{92054457}{1195357421}a^{5}+\frac{52962969}{1195357421}a^{4}+\frac{285689245}{1195357421}a^{3}+\frac{918678529}{2390714842}a^{2}-\frac{344218415}{2390714842}a+\frac{311690265}{2390714842}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$
Unit group
Rank: | $5$ | 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: | $\frac{13558}{4253941}a^{7}-\frac{30417}{8507882}a^{6}+\frac{113673}{4253941}a^{5}+\frac{623981}{4253941}a^{4}-\frac{13348015}{8507882}a^{3}-\frac{76291}{8507882}a^{2}+\frac{8829592}{4253941}a-\frac{6661999}{4253941}$, $\frac{3343157}{2390714842}a^{7}-\frac{16924713}{2390714842}a^{6}+\frac{71117875}{2390714842}a^{5}+\frac{7210087}{1195357421}a^{4}-\frac{924421587}{1195357421}a^{3}+\frac{4057130667}{1195357421}a^{2}-\frac{4600255859}{1195357421}a+\frac{2957373117}{2390714842}$, $\frac{22094861}{2390714842}a^{7}-\frac{15470198}{1195357421}a^{6}+\frac{103921548}{1195357421}a^{5}+\frac{1088608035}{2390714842}a^{4}-\frac{10773684199}{2390714842}a^{3}+\frac{1449253307}{1195357421}a^{2}+\frac{4767417145}{1195357421}a-\frac{8895934700}{1195357421}$, $\frac{76531839}{2390714842}a^{7}+\frac{14064594}{1195357421}a^{6}+\frac{275510013}{1195357421}a^{5}+\frac{4381981499}{2390714842}a^{4}-\frac{32129233885}{2390714842}a^{3}-\frac{26981523460}{1195357421}a^{2}+\frac{14266989109}{1195357421}a+\frac{24120764633}{1195357421}$, $\frac{939430}{1195357421}a^{7}+\frac{7526063}{2390714842}a^{6}+\frac{23522891}{1195357421}a^{5}+\frac{84709353}{1195357421}a^{4}+\frac{448138089}{2390714842}a^{3}-\frac{2672421219}{2390714842}a^{2}-\frac{1016074530}{1195357421}a+\frac{1213230474}{1195357421}$ | 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: | \( 3109.73065289 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{2}\cdot 3109.73065289 \cdot 1}{2\cdot\sqrt{20794740015625}}\cr\approx \mathstrut & 0.215375228270 \end{aligned}\]
Galois group
$C_2^2:C_4$ (as 8T10):
A solvable group of order 16 |
The 10 conjugacy class representatives for $C_2^2:C_4$ |
Character table for $C_2^2:C_4$ |
Intermediate fields
\(\Q(\sqrt{5}) \), 4.4.4560125.1, 4.2.23875.1, 4.2.4775.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
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 | ${\href{/padicField/7.4.0.1}{4} }^{2}$ | ${\href{/padicField/11.2.0.1}{2} }^{4}$ | ${\href{/padicField/13.4.0.1}{4} }^{2}$ | ${\href{/padicField/17.4.0.1}{4} }^{2}$ | ${\href{/padicField/19.2.0.1}{2} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }^{4}$ | ${\href{/padicField/23.4.0.1}{4} }^{2}$ | ${\href{/padicField/29.2.0.1}{2} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }^{4}$ | ${\href{/padicField/31.2.0.1}{2} }^{4}$ | ${\href{/padicField/37.4.0.1}{4} }^{2}$ | ${\href{/padicField/41.2.0.1}{2} }^{4}$ | ${\href{/padicField/43.4.0.1}{4} }^{2}$ | ${\href{/padicField/47.4.0.1}{4} }^{2}$ | ${\href{/padicField/53.4.0.1}{4} }^{2}$ | ${\href{/padicField/59.2.0.1}{2} }^{4}$ |
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 |
---|---|---|---|---|---|---|---|
\(5\) | 5.8.6.1 | $x^{8} + 16 x^{7} + 104 x^{6} + 352 x^{5} + 674 x^{4} + 784 x^{3} + 776 x^{2} + 928 x + 721$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ |
\(191\) | 191.4.2.1 | $x^{4} + 380 x^{3} + 36520 x^{2} + 79800 x + 6924684$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
191.4.2.1 | $x^{4} + 380 x^{3} + 36520 x^{2} + 79800 x + 6924684$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |