Normalized defining polynomial
\( x^{10} - 3x^{9} - 5x^{8} + 21x^{7} - 15x^{6} + 12x^{5} - 41x^{4} + 45x^{3} + 37x^{2} + 12x + 3 \)
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)
| |
Discriminant: | \(10059713635257\) \(\medspace = 3^{7}\cdot 11^{5}\cdot 13^{4}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(19.96\) | 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))
| |
Galois root discriminant: | $3^{3/4}11^{1/2}13^{2/3}\approx 41.79902823140501$ | ||
Ramified primes: | \(3\), \(11\), \(13\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{33}) \) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2685119}a^{9}+\frac{734917}{2685119}a^{8}-\frac{1114977}{2685119}a^{7}-\frac{1131589}{2685119}a^{6}-\frac{386572}{2685119}a^{5}-\frac{478433}{2685119}a^{4}+\frac{982411}{2685119}a^{3}-\frac{100388}{2685119}a^{2}-\frac{819279}{2685119}a+\frac{1191654}{2685119}$
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)
| |
Fundamental units: | $\frac{226572}{2685119}a^{9}-\frac{670023}{2685119}a^{8}-\frac{1203086}{2685119}a^{7}+\frac{4889926}{2685119}a^{6}-\frac{3179642}{2685119}a^{5}+\frac{1417473}{2685119}a^{4}-\frac{6854889}{2685119}a^{3}+\frac{8588470}{2685119}a^{2}+\frac{10020477}{2685119}a+\frac{1344400}{2685119}$, $\frac{226572}{2685119}a^{9}-\frac{670023}{2685119}a^{8}-\frac{1203086}{2685119}a^{7}+\frac{4889926}{2685119}a^{6}-\frac{3179642}{2685119}a^{5}+\frac{1417473}{2685119}a^{4}-\frac{6854889}{2685119}a^{3}+\frac{8588470}{2685119}a^{2}+\frac{7335358}{2685119}a+\frac{1344400}{2685119}$, $\frac{142031}{2685119}a^{9}-\frac{319579}{2685119}a^{8}-\frac{1035024}{2685119}a^{7}+\frac{2450724}{2685119}a^{6}+\frac{105580}{2685119}a^{5}-\frac{10890}{2685119}a^{4}-\frac{4762332}{2685119}a^{3}+\frac{2458981}{2685119}a^{2}+\frac{10041811}{2685119}a+\frac{6073585}{2685119}$, $\frac{987998}{2685119}a^{9}-\frac{2613338}{2685119}a^{8}-\frac{6180463}{2685119}a^{7}+\frac{19495279}{2685119}a^{6}-\frac{6406534}{2685119}a^{5}+\frac{2871864}{2685119}a^{4}-\frac{34304608}{2685119}a^{3}+\frac{29318707}{2685119}a^{2}+\frac{58292440}{2685119}a+\frac{18381238}{2685119}$, $\frac{228048}{2685119}a^{9}-\frac{720607}{2685119}a^{8}-\frac{931191}{2685119}a^{7}+\frac{4808580}{2685119}a^{6}-\frac{4514686}{2685119}a^{5}+\frac{4121781}{2685119}a^{4}-\frac{12150751}{2685119}a^{3}+\frac{16152684}{2685119}a^{2}+\frac{1012866}{2685119}a+\frac{1472759}{2685119}$ | 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: | \( 320.322969804 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{4}\cdot 320.322969804 \cdot 1}{2\cdot\sqrt{10059713635257}}\cr\approx \mathstrut & 0.314807262151 \end{aligned}\]
Galois group
A non-solvable group of order 120 |
The 7 conjugacy class representatives for $S_5$ |
Character table for $S_5$ |
Intermediate fields
\(\Q(\sqrt{33}) \), 5.1.50193.2 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 5 sibling: | 5.1.50193.2 |
Degree 6 sibling: | 6.2.1026396657.2 |
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: | 5.1.50193.2 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | ${\href{/padicField/2.5.0.1}{5} }^{2}$ | R | ${\href{/padicField/5.4.0.1}{4} }^{2}{,}\,{\href{/padicField/5.2.0.1}{2} }$ | ${\href{/padicField/7.4.0.1}{4} }^{2}{,}\,{\href{/padicField/7.2.0.1}{2} }$ | R | R | ${\href{/padicField/17.5.0.1}{5} }^{2}$ | ${\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.2.0.1}{2} }^{2}$ | ${\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.2.0.1}{2} }^{2}$ | ${\href{/padicField/29.5.0.1}{5} }^{2}$ | ${\href{/padicField/31.5.0.1}{5} }^{2}$ | ${\href{/padicField/37.5.0.1}{5} }^{2}$ | ${\href{/padicField/41.3.0.1}{3} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }^{4}$ | ${\href{/padicField/43.4.0.1}{4} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }$ | ${\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.2.0.1}{2} }^{2}$ | ${\href{/padicField/53.2.0.1}{2} }^{5}$ | ${\href{/padicField/59.4.0.1}{4} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }$ |
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 |
---|---|---|---|---|---|---|---|
\(3\) | 3.2.1.2 | $x^{2} + 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
3.8.6.2 | $x^{8} + 6 x^{5} + 6 x^{4} + 18 x^{2} + 18 x + 9$ | $4$ | $2$ | $6$ | $D_4$ | $[\ ]_{4}^{2}$ | |
\(11\) | 11.2.1.1 | $x^{2} + 22$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
11.4.2.1 | $x^{4} + 14 x^{3} + 75 x^{2} + 182 x + 620$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
11.4.2.1 | $x^{4} + 14 x^{3} + 75 x^{2} + 182 x + 620$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
\(13\) | 13.2.0.1 | $x^{2} + 12 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
13.2.0.1 | $x^{2} + 12 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
13.6.4.3 | $x^{6} + 36 x^{5} + 438 x^{4} + 1898 x^{3} + 1344 x^{2} + 5604 x + 21705$ | $3$ | $2$ | $4$ | $C_6$ | $[\ ]_{3}^{2}$ |