Normalized defining polynomial
\( x^{6} - 3x^{5} + 5x^{3} + 12x + 4 \)
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)
| |
Discriminant: | \(25312500\) \(\medspace = 2^{2}\cdot 3^{4}\cdot 5^{7}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(17.14\) | 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: | $2^{2/3}3^{4/5}5^{31/20}\approx 46.32198635146188$ | ||
Ramified primes: | \(2\), \(3\), \(5\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{5}) \) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{30}a^{5}+\frac{1}{6}a^{4}+\frac{1}{3}a^{3}-\frac{1}{6}a^{2}-\frac{1}{3}a-\frac{4}{15}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$
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: | $\frac{1}{10}a^{5}-\frac{1}{2}a^{4}+a^{3}-\frac{1}{2}a^{2}-2a+\frac{21}{5}$, $\frac{5}{6}a^{5}-\frac{17}{6}a^{4}+\frac{4}{3}a^{3}+\frac{17}{6}a^{2}-\frac{1}{3}a+\frac{31}{3}$, $\frac{4}{3}a^{5}-\frac{16}{3}a^{4}+\frac{4}{3}a^{3}+\frac{52}{3}a^{2}-\frac{37}{3}a-\frac{17}{3}$ | 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: | \( 95.4293841956 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{2}\cdot 95.4293841956 \cdot 1}{2\cdot\sqrt{25312500}}\cr\approx \mathstrut & 1.49762931833 \end{aligned}\]
Galois group
A non-solvable group of order 720 |
The 11 conjugacy class representatives for $S_6$ |
Character table for $S_6$ |
Intermediate fields
The extension is primitive: there are no intermediate fields between this field and $\Q$. |
Sibling algebras
Twin sextic algebra: | 6.2.101250000.1 |
Degree 6 sibling: | 6.2.101250000.1 |
Degree 10 sibling: | 10.2.12814453125000000.3 |
Degree 12 siblings: | deg 12, deg 12 |
Degree 15 siblings: | deg 15, deg 15 |
Degree 20 siblings: | deg 20, deg 20, deg 20 |
Degree 30 siblings: | deg 30, deg 30, deg 30, deg 30, deg 30, deg 30 |
Degree 36 sibling: | deg 36 |
Degree 40 siblings: | deg 40, deg 40, deg 40 |
Degree 45 sibling: | 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 | R | R | ${\href{/padicField/7.3.0.1}{3} }{,}\,{\href{/padicField/7.2.0.1}{2} }{,}\,{\href{/padicField/7.1.0.1}{1} }$ | ${\href{/padicField/11.4.0.1}{4} }{,}\,{\href{/padicField/11.2.0.1}{2} }$ | ${\href{/padicField/13.6.0.1}{6} }$ | ${\href{/padicField/17.2.0.1}{2} }^{3}$ | ${\href{/padicField/19.5.0.1}{5} }{,}\,{\href{/padicField/19.1.0.1}{1} }$ | ${\href{/padicField/23.3.0.1}{3} }{,}\,{\href{/padicField/23.2.0.1}{2} }{,}\,{\href{/padicField/23.1.0.1}{1} }$ | ${\href{/padicField/29.3.0.1}{3} }^{2}$ | ${\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.2.0.1}{2} }$ | ${\href{/padicField/37.3.0.1}{3} }{,}\,{\href{/padicField/37.2.0.1}{2} }{,}\,{\href{/padicField/37.1.0.1}{1} }$ | ${\href{/padicField/41.5.0.1}{5} }{,}\,{\href{/padicField/41.1.0.1}{1} }$ | ${\href{/padicField/43.6.0.1}{6} }$ | ${\href{/padicField/47.6.0.1}{6} }$ | ${\href{/padicField/53.3.0.1}{3} }{,}\,{\href{/padicField/53.2.0.1}{2} }{,}\,{\href{/padicField/53.1.0.1}{1} }$ | ${\href{/padicField/59.5.0.1}{5} }{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.3.2.1 | $x^{3} + 2$ | $3$ | $1$ | $2$ | $S_3$ | $[\ ]_{3}^{2}$ |
2.3.0.1 | $x^{3} + x + 1$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
\(3\) | $\Q_{3}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
3.5.4.1 | $x^{5} + 3$ | $5$ | $1$ | $4$ | $F_5$ | $[\ ]_{5}^{4}$ | |
\(5\) | $\Q_{5}$ | $x + 3$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
5.5.7.2 | $x^{5} + 10 x^{3} + 5$ | $5$ | $1$ | $7$ | $F_5$ | $[7/4]_{4}$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
1.5.2t1.a.a | $1$ | $ 5 $ | \(\Q(\sqrt{5}) \) | $C_2$ (as 2T1) | $1$ | $1$ | |
* | 5.25312500.6t16.a.a | $5$ | $ 2^{2} \cdot 3^{4} \cdot 5^{7}$ | 6.2.25312500.1 | $S_6$ (as 6T16) | $1$ | $1$ |
5.506250000.12t183.a.a | $5$ | $ 2^{4} \cdot 3^{4} \cdot 5^{8}$ | 6.2.25312500.1 | $S_6$ (as 6T16) | $1$ | $1$ | |
5.126562500.12t183.a.a | $5$ | $ 2^{2} \cdot 3^{4} \cdot 5^{8}$ | 6.2.25312500.1 | $S_6$ (as 6T16) | $1$ | $1$ | |
5.101250000.6t16.a.a | $5$ | $ 2^{4} \cdot 3^{4} \cdot 5^{7}$ | 6.2.25312500.1 | $S_6$ (as 6T16) | $1$ | $1$ | |
9.128...000.10t32.a.a | $9$ | $ 2^{6} \cdot 3^{8} \cdot 5^{15}$ | 6.2.25312500.1 | $S_6$ (as 6T16) | $1$ | $1$ | |
9.256...000.20t145.a.a | $9$ | $ 2^{6} \cdot 3^{8} \cdot 5^{14}$ | 6.2.25312500.1 | $S_6$ (as 6T16) | $1$ | $1$ | |
10.640...000.30t164.a.a | $10$ | $ 2^{6} \cdot 3^{8} \cdot 5^{16}$ | 6.2.25312500.1 | $S_6$ (as 6T16) | $1$ | $-2$ | |
10.640...000.30t164.b.a | $10$ | $ 2^{6} \cdot 3^{8} \cdot 5^{16}$ | 6.2.25312500.1 | $S_6$ (as 6T16) | $1$ | $-2$ | |
16.129...000.36t1252.a.a | $16$ | $ 2^{12} \cdot 3^{12} \cdot 5^{24}$ | 6.2.25312500.1 | $S_6$ (as 6T16) | $1$ | $0$ |