Normalized defining polynomial
\( x^{6} - 3x^{5} + 24x^{4} + 81x^{3} + 44x^{2} + 1217x + 1831 \)
Invariants
Degree: | $6$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[2, 2]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(36940840000\) \(\medspace = 2^{6}\cdot 5^{4}\cdot 31^{4}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(57.71\) | 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: | $2^{3/2}5^{3/4}31^{2/3}\approx 93.32835820701949$ | ||
Ramified primes: | \(2\), \(5\), \(31\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q\) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{3}-\frac{1}{2}$, $\frac{1}{4}a^{4}-\frac{1}{4}a^{2}-\frac{1}{2}a-\frac{1}{4}$, $\frac{1}{15132}a^{5}+\frac{85}{3783}a^{4}-\frac{629}{15132}a^{3}+\frac{937}{3783}a^{2}-\frac{1465}{5044}a-\frac{317}{7566}$
Monogenic: | No | |
Index: | $2$ | |
Inessential primes: | $2$ |
Class group and class number
$C_{3}\times C_{3}$, which has order $9$
Unit group
Rank: | $3$ | 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: | $\frac{11}{15132}a^{5}-\frac{43}{15132}a^{4}+\frac{647}{15132}a^{3}-\frac{385}{15132}a^{2}+\frac{1539}{5044}a+\frac{34639}{15132}$, $\frac{589}{15132}a^{5}-\frac{239}{15132}a^{4}+\frac{253}{15132}a^{3}+\frac{17215}{15132}a^{2}-\frac{12971}{5044}a-\frac{89701}{15132}$, $\frac{6350}{3783}a^{5}-\frac{121645}{15132}a^{4}+\frac{212546}{3783}a^{3}+\frac{938189}{15132}a^{2}+\frac{10643}{2522}a+\frac{34494025}{15132}$ | 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: | \( 189.748143705 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{2}\cdot 189.748143705 \cdot 9}{2\cdot\sqrt{36940840000}}\cr\approx \mathstrut & 0.701546390339 \end{aligned}\]
Galois group
$C_3^2:C_4$ (as 6T10):
A solvable group of order 36 |
The 6 conjugacy class representatives for $C_3^2:C_4$ |
Character table for $C_3^2:C_4$ |
Intermediate fields
\(\Q(\sqrt{5}) \) |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling algebras
Galois closure: | deg 36 |
Twin sextic algebra: | 6.2.38440000.1 |
Degree 6 sibling: | 6.2.38440000.1 |
Degree 9 sibling: | 9.1.56800235584000000.1 |
Degree 12 siblings: | deg 12, deg 12 |
Degree 18 sibling: | deg 18 |
Minimal sibling: | 6.2.38440000.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | ${\href{/padicField/3.4.0.1}{4} }{,}\,{\href{/padicField/3.2.0.1}{2} }$ | R | ${\href{/padicField/7.4.0.1}{4} }{,}\,{\href{/padicField/7.2.0.1}{2} }$ | ${\href{/padicField/11.2.0.1}{2} }^{2}{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ | ${\href{/padicField/13.4.0.1}{4} }{,}\,{\href{/padicField/13.2.0.1}{2} }$ | ${\href{/padicField/17.4.0.1}{4} }{,}\,{\href{/padicField/17.2.0.1}{2} }$ | ${\href{/padicField/19.3.0.1}{3} }{,}\,{\href{/padicField/19.1.0.1}{1} }^{3}$ | ${\href{/padicField/23.4.0.1}{4} }{,}\,{\href{/padicField/23.2.0.1}{2} }$ | ${\href{/padicField/29.3.0.1}{3} }^{2}$ | R | ${\href{/padicField/37.4.0.1}{4} }{,}\,{\href{/padicField/37.2.0.1}{2} }$ | ${\href{/padicField/41.3.0.1}{3} }^{2}$ | ${\href{/padicField/43.4.0.1}{4} }{,}\,{\href{/padicField/43.2.0.1}{2} }$ | ${\href{/padicField/47.4.0.1}{4} }{,}\,{\href{/padicField/47.2.0.1}{2} }$ | ${\href{/padicField/53.4.0.1}{4} }{,}\,{\href{/padicField/53.2.0.1}{2} }$ | ${\href{/padicField/59.3.0.1}{3} }{,}\,{\href{/padicField/59.1.0.1}{1} }^{3}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.2.0.1 | $x^{2} + x + 1$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
2.4.6.3 | $x^{4} + 8 x^{3} + 28 x^{2} + 48 x + 84$ | $2$ | $2$ | $6$ | $C_4$ | $[3]^{2}$ | |
\(5\) | 5.2.1.1 | $x^{2} + 5$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
5.4.3.1 | $x^{4} + 20$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ | |
\(31\) | 31.3.2.3 | $x^{3} + 155$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ |
31.3.2.3 | $x^{3} + 155$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ |