Normalized defining polynomial
\( x^{6} - 3x^{5} + 67x^{4} - 7x^{3} + 214x^{2} - 111x + 225 \)
Invariants
Degree: | $6$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[0, 3]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(-3315280740031\) \(\medspace = -\,13^{3}\cdot 31^{3}\cdot 37^{3}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(122.11\) | 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: | $13^{1/2}31^{1/2}37^{1/2}\approx 122.11060560000512$ | ||
Ramified primes: | \(13\), \(31\), \(37\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{-14911}) \) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{9261}a^{5}+\frac{76}{441}a^{4}-\frac{3965}{9261}a^{3}+\frac{3743}{9261}a^{2}+\frac{2665}{9261}a+\frac{388}{3087}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{2}\times C_{2}\times C_{4}$, which has order $16$
Unit group
Rank: | $2$ | 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: | $\frac{2384}{147}a^{5}-\frac{123}{7}a^{4}+\frac{134339}{147}a^{3}+\frac{322636}{147}a^{2}-\frac{262522}{147}a+\frac{177007}{49}$, $\frac{6726952849}{343}a^{5}-\frac{2850397441}{49}a^{4}+\frac{429897545655}{343}a^{3}+\frac{48032203273}{343}a^{2}+\frac{71290617851}{343}a+\frac{529560376172}{343}$ | 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: | \( 1386.32837273 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{3}\cdot 1386.32837273 \cdot 16}{2\cdot\sqrt{3315280740031}}\cr\approx \mathstrut & 1.51089939883 \end{aligned}\]
Galois group
A non-solvable group of order 120 |
The 7 conjugacy class representatives for $\PGL(2,5)$ |
Character table for $\PGL(2,5)$ |
Intermediate fields
The extension is primitive: there are no intermediate fields between this field and $\Q$. |
Sibling algebras
Twin sextic algebra: | \(\Q\) $\times$ 5.3.14911.1 |
Degree 5 sibling: | 5.3.14911.1 |
Degree 10 siblings: | 10.4.3315280740031.1, deg 10 |
Degree 12 sibling: | deg 12 |
Degree 15 sibling: | deg 15 |
Degree 20 siblings: | deg 20, deg 20, deg 20 |
Degree 24 sibling: | deg 24 |
Degree 30 siblings: | deg 30, deg 30, deg 30 |
Degree 40 sibling: | deg 40 |
Minimal sibling: | 5.3.14911.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | ${\href{/padicField/2.5.0.1}{5} }{,}\,{\href{/padicField/2.1.0.1}{1} }$ | ${\href{/padicField/3.4.0.1}{4} }{,}\,{\href{/padicField/3.1.0.1}{1} }^{2}$ | ${\href{/padicField/5.5.0.1}{5} }{,}\,{\href{/padicField/5.1.0.1}{1} }$ | ${\href{/padicField/7.4.0.1}{4} }{,}\,{\href{/padicField/7.1.0.1}{1} }^{2}$ | ${\href{/padicField/11.5.0.1}{5} }{,}\,{\href{/padicField/11.1.0.1}{1} }$ | R | ${\href{/padicField/17.5.0.1}{5} }{,}\,{\href{/padicField/17.1.0.1}{1} }$ | ${\href{/padicField/19.2.0.1}{2} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | ${\href{/padicField/23.5.0.1}{5} }{,}\,{\href{/padicField/23.1.0.1}{1} }$ | ${\href{/padicField/29.5.0.1}{5} }{,}\,{\href{/padicField/29.1.0.1}{1} }$ | R | R | ${\href{/padicField/41.2.0.1}{2} }^{3}$ | ${\href{/padicField/43.5.0.1}{5} }{,}\,{\href{/padicField/43.1.0.1}{1} }$ | ${\href{/padicField/47.6.0.1}{6} }$ | ${\href{/padicField/53.4.0.1}{4} }{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | ${\href{/padicField/59.5.0.1}{5} }{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 |
---|---|---|---|---|---|---|---|
\(13\) | 13.6.3.1 | $x^{6} + 390 x^{5} + 50743 x^{4} + 2208202 x^{3} + 765301 x^{2} + 5017316 x + 24555184$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ |
\(31\) | 31.2.1.2 | $x^{2} + 31$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
31.4.2.1 | $x^{4} + 58 x^{3} + 909 x^{2} + 1972 x + 26855$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
\(37\) | 37.2.1.1 | $x^{2} + 37$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
37.4.2.1 | $x^{4} + 1916 x^{3} + 948367 x^{2} + 29317674 x + 2943243$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |