Normalized defining polynomial
\( x^{10} + 5x^{6} - 33x^{5} + 75x^{4} - 115x^{3} + 110x^{2} - 55x + 11 \)
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: | \(714892578125\) \(\medspace = 5^{11}\cdot 11^{4}\) | 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: | \(15.33\) | 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: | $5^{23/20}11^{2/3}\approx 31.483085955652367$ | ||
Ramified primes: | \(5\), \(11\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{5}) \) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{15282}a^{9}-\frac{2035}{15282}a^{8}+\frac{3722}{7641}a^{7}+\frac{3563}{15282}a^{6}-\frac{1172}{2547}a^{5}-\frac{251}{2547}a^{4}-\frac{2299}{5094}a^{3}+\frac{3202}{7641}a^{2}-\frac{1375}{5094}a+\frac{2251}{7641}$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $2$ |
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: | $\frac{803}{15282}a^{9}+\frac{1069}{15282}a^{8}+\frac{1135}{7641}a^{7}+\frac{3355}{15282}a^{6}+\frac{1274}{2547}a^{5}-\frac{2887}{2547}a^{4}+\frac{13213}{5094}a^{3}-\frac{34375}{7641}a^{2}+\frac{16555}{5094}a-\frac{18646}{7641}$, $\frac{17047}{7641}a^{9}+\frac{14777}{7641}a^{8}+\frac{11422}{7641}a^{7}+\frac{7793}{7641}a^{6}+\frac{29732}{2547}a^{5}-\frac{162682}{2547}a^{4}+\frac{282353}{2547}a^{3}-\frac{1189975}{7641}a^{2}+\frac{265304}{2547}a-\frac{168712}{7641}$, $a-1$, $\frac{10010}{7641}a^{9}+\frac{8197}{7641}a^{8}+\frac{7049}{7641}a^{7}+\frac{5083}{7641}a^{6}+\frac{17353}{2547}a^{5}-\frac{96575}{2547}a^{4}+\frac{172351}{2547}a^{3}-\frac{737486}{7641}a^{2}+\frac{178528}{2547}a-\frac{146777}{7641}$, $\frac{26462}{7641}a^{9}+\frac{18880}{7641}a^{8}+\frac{13430}{7641}a^{7}+\frac{9448}{7641}a^{6}+\frac{46009}{2547}a^{5}-\frac{258566}{2547}a^{4}+\frac{477793}{2547}a^{3}-\frac{2016674}{7641}a^{2}+\frac{487669}{2547}a-\frac{396239}{7641}$ | 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: | \( 140.686816299 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{4}\cdot 140.686816299 \cdot 1}{2\cdot\sqrt{714892578125}}\cr\approx \mathstrut & 0.518659666823 \end{aligned}\]
Galois group
A non-solvable group of order 120 |
The 7 conjugacy class representatives for $S_5$ |
Character table for $S_5$ |
Intermediate fields
\(\Q(\sqrt{5}) \), 5.1.378125.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 5 sibling: | 5.1.378125.1 |
Degree 6 sibling: | 6.2.45753125.1 |
Degree 10 sibling: | data not computed |
Degree 12 sibling: | data not computed |
Degree 15 sibling: | data not computed |
Degree 20 siblings: | data not computed |
Degree 24 sibling: | data not computed |
Degree 30 siblings: | data not computed |
Degree 40 sibling: | data not computed |
Minimal sibling: | 5.1.378125.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | ${\href{/padicField/2.6.0.1}{6} }{,}\,{\href{/padicField/2.2.0.1}{2} }^{2}$ | ${\href{/padicField/3.6.0.1}{6} }{,}\,{\href{/padicField/3.2.0.1}{2} }^{2}$ | R | ${\href{/padicField/7.4.0.1}{4} }^{2}{,}\,{\href{/padicField/7.2.0.1}{2} }$ | R | ${\href{/padicField/13.4.0.1}{4} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }$ | ${\href{/padicField/17.2.0.1}{2} }^{5}$ | ${\href{/padicField/19.5.0.1}{5} }^{2}$ | ${\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.2.0.1}{2} }^{2}$ | ${\href{/padicField/29.5.0.1}{5} }^{2}$ | ${\href{/padicField/31.5.0.1}{5} }^{2}$ | ${\href{/padicField/37.4.0.1}{4} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }$ | ${\href{/padicField/41.2.0.1}{2} }^{4}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.2.0.1}{2} }^{5}$ | ${\href{/padicField/47.4.0.1}{4} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }$ | ${\href{/padicField/53.4.0.1}{4} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }$ | ${\href{/padicField/59.3.0.1}{3} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }^{4}$ |
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 |
---|---|---|---|---|---|---|---|
\(5\) | 5.10.11.2 | $x^{10} + 5 x^{2} + 5$ | $10$ | $1$ | $11$ | $F_5$ | $[5/4]_{4}$ |
\(11\) | 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 | $x^{3} + 11$ | $3$ | $1$ | $2$ | $S_3$ | $[\ ]_{3}^{2}$ |