Normalized defining polynomial
\( x^{6} - 2x^{5} - 39x^{4} + 59x^{3} + 381x^{2} - 380x - 880 \)
Invariants
Degree: | $6$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[6, 0]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(292281234205\) \(\medspace = 5\cdot 3881^{3}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(81.46\) | 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: | $5^{1/2}3881^{1/2}\approx 139.30183056945089$ | ||
Ramified primes: | \(5\), \(3881\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{19405}) \) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{2476}a^{5}+\frac{71}{1238}a^{4}+\frac{601}{2476}a^{3}-\frac{57}{2476}a^{2}-\frac{399}{2476}a-\frac{222}{619}$
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: | $\frac{105}{2476}a^{5}+\frac{27}{1238}a^{4}-\frac{3747}{2476}a^{3}-\frac{3509}{2476}a^{2}+\frac{24957}{2476}a+\frac{7021}{619}$, $\frac{4}{619}a^{5}-\frac{51}{619}a^{4}-\frac{72}{619}a^{3}+\frac{1629}{619}a^{2}-\frac{358}{619}a-\frac{9123}{619}$, $\frac{43}{1238}a^{5}-\frac{42}{619}a^{4}-\frac{1393}{1238}a^{3}+\frac{1263}{1238}a^{2}+\frac{7603}{1238}a+\frac{2573}{619}$, $\frac{335}{2476}a^{5}+\frac{263}{1238}a^{4}-\frac{11601}{2476}a^{3}-\frac{21571}{2476}a^{2}+\frac{66891}{2476}a+\frac{24051}{619}$, $\frac{687845189}{2476}a^{5}+\frac{201165735}{1238}a^{4}-\frac{24611023575}{2476}a^{3}-\frac{26930637909}{2476}a^{2}+\frac{171677806741}{2476}a+\frac{52844501537}{619}$ | 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: | \( 22465.3120857 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{6}\cdot(2\pi)^{0}\cdot 22465.3120857 \cdot 1}{2\cdot\sqrt{292281234205}}\cr\approx \mathstrut & 1.32972541100 \end{aligned}\]
Galois group
$\SOPlus(4,2)$ (as 6T13):
A solvable group of order 72 |
The 9 conjugacy class representatives for $C_3^2:D_4$ |
Character table for $C_3^2:D_4$ |
Intermediate fields
\(\Q(\sqrt{3881}) \) |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling algebras
Twin sextic algebra: | 6.6.485125.1 |
Degree 6 sibling: | 6.6.485125.1 |
Degree 9 sibling: | 9.9.7307030855125.1 |
Degree 12 siblings: | deg 12, 12.12.913378856890625.1, deg 12, deg 12, deg 12, deg 12 |
Degree 18 siblings: | 18.18.6674087489718598592345703125.1, deg 18, deg 18 |
Degree 24 siblings: | deg 24, deg 24 |
Degree 36 siblings: | deg 36, deg 36, deg 36 |
Minimal sibling: | 6.6.485125.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | ${\href{/padicField/2.3.0.1}{3} }{,}\,{\href{/padicField/2.2.0.1}{2} }{,}\,{\href{/padicField/2.1.0.1}{1} }$ | ${\href{/padicField/3.4.0.1}{4} }{,}\,{\href{/padicField/3.2.0.1}{2} }$ | R | ${\href{/padicField/7.4.0.1}{4} }{,}\,{\href{/padicField/7.2.0.1}{2} }$ | ${\href{/padicField/11.3.0.1}{3} }{,}\,{\href{/padicField/11.1.0.1}{1} }^{3}$ | ${\href{/padicField/13.4.0.1}{4} }{,}\,{\href{/padicField/13.2.0.1}{2} }$ | ${\href{/padicField/17.4.0.1}{4} }{,}\,{\href{/padicField/17.2.0.1}{2} }$ | ${\href{/padicField/19.2.0.1}{2} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | ${\href{/padicField/23.4.0.1}{4} }{,}\,{\href{/padicField/23.2.0.1}{2} }$ | ${\href{/padicField/29.2.0.1}{2} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.6.0.1}{6} }$ | ${\href{/padicField/37.3.0.1}{3} }{,}\,{\href{/padicField/37.2.0.1}{2} }{,}\,{\href{/padicField/37.1.0.1}{1} }$ | ${\href{/padicField/41.6.0.1}{6} }$ | ${\href{/padicField/43.2.0.1}{2} }{,}\,{\href{/padicField/43.1.0.1}{1} }^{4}$ | ${\href{/padicField/47.3.0.1}{3} }{,}\,{\href{/padicField/47.2.0.1}{2} }{,}\,{\href{/padicField/47.1.0.1}{1} }$ | ${\href{/padicField/53.4.0.1}{4} }{,}\,{\href{/padicField/53.2.0.1}{2} }$ | ${\href{/padicField/59.3.0.1}{3} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(5\) | $\Q_{5}$ | $x + 3$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
5.2.1.1 | $x^{2} + 5$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
5.3.0.1 | $x^{3} + 3 x + 3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
\(3881\) | Deg $6$ | $2$ | $3$ | $3$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
1.19405.2t1.a.a | $1$ | $ 5 \cdot 3881 $ | \(\Q(\sqrt{19405}) \) | $C_2$ (as 2T1) | $1$ | $1$ | |
* | 1.3881.2t1.a.a | $1$ | $ 3881 $ | \(\Q(\sqrt{3881}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
1.5.2t1.a.a | $1$ | $ 5 $ | \(\Q(\sqrt{5}) \) | $C_2$ (as 2T1) | $1$ | $1$ | |
2.19405.4t3.b.a | $2$ | $ 5 \cdot 3881 $ | 4.4.97025.1 | $D_{4}$ (as 4T3) | $1$ | $2$ | |
4.1882770125.12t34.a.a | $4$ | $ 5^{3} \cdot 3881^{2}$ | 6.6.292281234205.1 | $C_3^2:D_4$ (as 6T13) | $1$ | $4$ | |
* | 4.75310805.6t13.a.a | $4$ | $ 5 \cdot 3881^{2}$ | 6.6.292281234205.1 | $C_3^2:D_4$ (as 6T13) | $1$ | $4$ |
4.97025.6t13.a.a | $4$ | $ 5^{2} \cdot 3881 $ | 6.6.292281234205.1 | $C_3^2:D_4$ (as 6T13) | $1$ | $4$ | |
4.146...025.12t34.a.a | $4$ | $ 5^{2} \cdot 3881^{3}$ | 6.6.292281234205.1 | $C_3^2:D_4$ (as 6T13) | $1$ | $4$ |