Normalized defining polynomial
\( x^{10} - 4x^{9} + 11x^{8} - 20x^{7} + 73x^{5} - 227x^{4} + 454x^{3} - 515x^{2} + 445x - 271 \)
Invariants
Degree: | $10$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[2, 4]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(1015264874437\) \(\medspace = 11^{4}\cdot 37^{5}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(15.87\) | 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: | $11^{2/3}37^{1/2}\approx 30.08587537137172$ | ||
Ramified primes: | \(11\), \(37\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{37}) \) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{3506347}a^{9}-\frac{221816}{3506347}a^{8}+\frac{389499}{3506347}a^{7}+\frac{837872}{3506347}a^{6}+\frac{352324}{3506347}a^{5}-\frac{229079}{3506347}a^{4}-\frac{1509803}{3506347}a^{3}+\frac{1221520}{3506347}a^{2}+\frac{1663323}{3506347}a-\frac{156797}{3506347}$
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)
| |
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: | $\frac{2923}{3506347}a^{9}+\frac{306027}{3506347}a^{8}-\frac{1057198}{3506347}a^{7}+\frac{1669650}{3506347}a^{6}-\frac{1022966}{3506347}a^{5}-\frac{6898334}{3506347}a^{4}+\frac{22374786}{3506347}a^{3}-\frac{30515409}{3506347}a^{2}+\frac{33653310}{3506347}a-\frac{20024256}{3506347}$, $\frac{98908}{3506347}a^{9}-\frac{163749}{3506347}a^{8}+\frac{332603}{3506347}a^{7}-\frac{267569}{3506347}a^{6}-\frac{1920641}{3506347}a^{5}+\frac{3774929}{3506347}a^{4}-\frac{6795435}{3506347}a^{3}+\frac{10420622}{3506347}a^{2}-\frac{1849956}{3506347}a+\frac{95105}{3506347}$, $\frac{176546}{3506347}a^{9}-\frac{1844240}{3506347}a^{8}+\frac{5025784}{3506347}a^{7}-\frac{6323471}{3506347}a^{6}-\frac{1202876}{3506347}a^{5}+\frac{41301328}{3506347}a^{4}-\frac{102371908}{3506347}a^{3}+\frac{126332524}{3506347}a^{2}-\frac{122754690}{3506347}a+\frac{67346996}{3506347}$, $\frac{36990}{3506347}a^{9}-\frac{121860}{3506347}a^{8}-\frac{11813}{3506347}a^{7}+\frac{284147}{3506347}a^{6}-\frac{627039}{3506347}a^{5}+\frac{1208489}{3506347}a^{4}+\frac{1482046}{3506347}a^{3}-\frac{2268989}{3506347}a^{2}-\frac{6565733}{3506347}a+\frac{6589602}{3506347}$, $\frac{61858}{3506347}a^{9}-\frac{758317}{3506347}a^{8}+\frac{1518905}{3506347}a^{7}-\frac{1735178}{3506347}a^{6}-\frac{1394960}{3506347}a^{5}+\frac{16311180}{3506347}a^{4}-\frac{29892405}{3506347}a^{3}+\frac{34069780}{3506347}a^{2}-\frac{31969357}{3506347}a+\frac{9926017}{3506347}$ | 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: | \( 106.466480264 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{4}\cdot 106.466480264 \cdot 1}{2\cdot\sqrt{1015264874437}}\cr\approx \mathstrut & 0.329361388853 \end{aligned}\]
Galois group
A non-solvable group of order 120 |
The 7 conjugacy class representatives for $S_5$ |
Character table for $S_5$ |
Intermediate fields
\(\Q(\sqrt{37}) \), 5.1.4477.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 5 sibling: | 5.1.4477.1 |
Degree 6 sibling: | 6.2.741610573.1 |
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: | 5.1.4477.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | ${\href{/padicField/2.4.0.1}{4} }^{2}{,}\,{\href{/padicField/2.2.0.1}{2} }$ | ${\href{/padicField/3.5.0.1}{5} }^{2}$ | ${\href{/padicField/5.6.0.1}{6} }{,}\,{\href{/padicField/5.2.0.1}{2} }^{2}$ | ${\href{/padicField/7.5.0.1}{5} }^{2}$ | R | ${\href{/padicField/13.6.0.1}{6} }{,}\,{\href{/padicField/13.2.0.1}{2} }^{2}$ | ${\href{/padicField/17.4.0.1}{4} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }$ | ${\href{/padicField/19.4.0.1}{4} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }$ | ${\href{/padicField/23.4.0.1}{4} }^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }$ | ${\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.2.0.1}{2} }^{2}$ | ${\href{/padicField/31.4.0.1}{4} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }$ | R | ${\href{/padicField/41.3.0.1}{3} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }^{4}$ | ${\href{/padicField/43.4.0.1}{4} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }$ | ${\href{/padicField/47.5.0.1}{5} }^{2}$ | ${\href{/padicField/53.3.0.1}{3} }^{2}{,}\,{\href{/padicField/53.1.0.1}{1} }^{4}$ | ${\href{/padicField/59.4.0.1}{4} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }$ |
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 |
---|---|---|---|---|---|---|---|
\(11\) | 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 | $x^{3} + 11$ | $3$ | $1$ | $2$ | $S_3$ | $[\ ]_{3}^{2}$ | |
\(37\) | 37.2.1.1 | $x^{2} + 37$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
37.4.2.1 | $x^{4} + 1916 x^{3} + 948367 x^{2} + 29317674 x + 2943243$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
37.4.2.1 | $x^{4} + 1916 x^{3} + 948367 x^{2} + 29317674 x + 2943243$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |