Normalized defining polynomial
\( x^{10} - 5x^{9} + 16x^{8} - 34x^{7} + 38x^{6} - 16x^{5} - 72x^{4} + 141x^{3} - 175x^{2} + 106x - 25 \)
Invariants
Degree: | $10$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[2, 4]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(38194822598416\) \(\medspace = 2^{4}\cdot 11^{4}\cdot 113^{4}\) | 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: | \(22.81\) | 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: | $2^{2/3}11^{2/3}113^{2/3}\approx 183.5132456094119$ | ||
Ramified primes: | \(2\), \(11\), \(113\) | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{14}a^{8}-\frac{2}{7}a^{7}-\frac{1}{14}a^{6}+\frac{3}{14}a^{5}-\frac{1}{7}a^{4}-\frac{1}{14}a^{3}-\frac{5}{14}a^{2}-\frac{5}{14}a-\frac{3}{14}$, $\frac{1}{742}a^{9}+\frac{11}{371}a^{8}+\frac{19}{106}a^{7}+\frac{271}{742}a^{6}-\frac{165}{371}a^{5}+\frac{31}{742}a^{4}+\frac{235}{742}a^{3}+\frac{285}{742}a^{2}+\frac{37}{106}a-\frac{81}{371}$
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: | $\frac{36}{371}a^{9}-\frac{162}{371}a^{8}+\frac{442}{371}a^{7}-\frac{113}{53}a^{6}+\frac{67}{53}a^{5}+\frac{61}{53}a^{4}-\frac{2458}{371}a^{3}+\frac{3158}{371}a^{2}-\frac{2601}{371}a+\frac{1111}{371}$, $\frac{36}{371}a^{9}-\frac{162}{371}a^{8}+\frac{442}{371}a^{7}-\frac{113}{53}a^{6}+\frac{67}{53}a^{5}+\frac{61}{53}a^{4}-\frac{2458}{371}a^{3}+\frac{2787}{371}a^{2}-\frac{2230}{371}a-\frac{2}{371}$, $\frac{18}{371}a^{9}-\frac{4}{53}a^{8}+\frac{9}{371}a^{7}+\frac{108}{371}a^{6}-\frac{534}{371}a^{5}+\frac{293}{371}a^{4}-\frac{169}{371}a^{3}-\frac{1283}{371}a^{2}+\frac{1588}{371}a-\frac{531}{371}$, $\frac{194}{371}a^{9}-\frac{873}{371}a^{8}+\frac{2588}{371}a^{7}-\frac{712}{53}a^{6}+\frac{576}{53}a^{5}+\frac{49}{53}a^{4}-\frac{14194}{371}a^{3}+\frac{19038}{371}a^{2}-\frac{19396}{371}a+\frac{6626}{371}$, $a^{2}-a+2$ | 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: | \( 1001.32800374 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
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Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{4}\cdot 1001.32800374 \cdot 1}{2\cdot\sqrt{38194822598416}}\cr\approx \mathstrut & 0.505037448890 \end{aligned}\]
Galois group
$C_2^4:A_5$ (as 10T34):
A non-solvable group of order 960 |
The 12 conjugacy class representatives for $C_2^4 : A_5$ |
Character table for $C_2^4 : A_5$ |
Intermediate fields
5.5.6180196.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 16 sibling: | data not computed |
Degree 20 siblings: | data not computed |
Degree 30 siblings: | data not computed |
Degree 40 siblings: | 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 | R | ${\href{/padicField/3.5.0.1}{5} }^{2}$ | ${\href{/padicField/5.6.0.1}{6} }{,}\,{\href{/padicField/5.2.0.1}{2} }{,}\,{\href{/padicField/5.1.0.1}{1} }^{2}$ | ${\href{/padicField/7.3.0.1}{3} }^{2}{,}\,{\href{/padicField/7.2.0.1}{2} }^{2}$ | R | ${\href{/padicField/13.5.0.1}{5} }^{2}$ | ${\href{/padicField/17.5.0.1}{5} }^{2}$ | ${\href{/padicField/19.5.0.1}{5} }^{2}$ | ${\href{/padicField/23.4.0.1}{4} }^{2}{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ | ${\href{/padicField/29.3.0.1}{3} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }^{4}$ | ${\href{/padicField/31.5.0.1}{5} }^{2}$ | ${\href{/padicField/37.4.0.1}{4} }{,}\,{\href{/padicField/37.2.0.1}{2} }^{3}$ | ${\href{/padicField/41.5.0.1}{5} }^{2}$ | ${\href{/padicField/43.5.0.1}{5} }^{2}$ | ${\href{/padicField/47.5.0.1}{5} }^{2}$ | ${\href{/padicField/53.4.0.1}{4} }^{2}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | ${\href{/padicField/59.5.0.1}{5} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.4.0.1 | $x^{4} + x + 1$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |
2.6.4.1 | $x^{6} + 3 x^{5} + 10 x^{4} + 19 x^{3} + 22 x^{2} + 11 x + 7$ | $3$ | $2$ | $4$ | $S_3$ | $[\ ]_{3}^{2}$ | |
\(11\) | 11.2.0.1 | $x^{2} + 7 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
11.2.0.1 | $x^{2} + 7 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
11.3.2.1 | $x^{3} + 11$ | $3$ | $1$ | $2$ | $S_3$ | $[\ ]_{3}^{2}$ | |
11.3.2.1 | $x^{3} + 11$ | $3$ | $1$ | $2$ | $S_3$ | $[\ ]_{3}^{2}$ | |
\(113\) | 113.2.0.1 | $x^{2} + 101 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
113.2.0.1 | $x^{2} + 101 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
113.3.2.1 | $x^{3} + 113$ | $3$ | $1$ | $2$ | $S_3$ | $[\ ]_{3}^{2}$ | |
113.3.2.1 | $x^{3} + 113$ | $3$ | $1$ | $2$ | $S_3$ | $[\ ]_{3}^{2}$ |