Normalized defining polynomial
\( x^{6} - x^{5} - 321x^{4} + 321x^{3} + 29303x^{2} - 29303x - 652049 \)
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)
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Discriminant: | \(103001481009\) \(\medspace = 3^{3}\cdot 7^{5}\cdot 61^{3}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(68.47\) | 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: | $3^{1/2}7^{5/6}61^{1/2}\approx 68.46583538188537$ | ||
Ramified primes: | \(3\), \(7\), \(61\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{1281}) \) | ||
$\card{ \Gal(K/\Q) }$: | $6$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
This field is Galois and abelian over $\Q$. | |||
Conductor: | \(1281=3\cdot 7\cdot 61\) | ||
Dirichlet character group: | $\lbrace$$\chi_{1281}(1280,·)$, $\chi_{1281}(1,·)$, $\chi_{1281}(550,·)$, $\chi_{1281}(184,·)$, $\chi_{1281}(1097,·)$, $\chi_{1281}(731,·)$$\rbrace$ | ||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{84869}a^{4}-\frac{1888}{84869}a^{3}-\frac{184}{84869}a^{2}+\frac{5937}{84869}a+\frac{4232}{84869}$, $\frac{1}{84869}a^{5}-\frac{230}{84869}a^{3}-\frac{1979}{84869}a^{2}+\frac{10580}{84869}a+\frac{12330}{84869}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{2}$, which has order $2$
Unit group
Rank: | $5$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
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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: | $\frac{45}{84869}a^{4}-\frac{91}{84869}a^{3}-\frac{8280}{84869}a^{2}+\frac{12558}{84869}a+\frac{190440}{84869}$, $\frac{1}{84869}a^{5}-\frac{230}{84869}a^{3}-\frac{1979}{84869}a^{2}+\frac{10580}{84869}a+\frac{182068}{84869}$, $\frac{490}{84869}a^{5}-\frac{3568}{84869}a^{4}-\frac{80967}{84869}a^{3}+\frac{450623}{84869}a^{2}+\frac{1823474}{84869}a-\frac{7360696}{84869}$, $\frac{606}{84869}a^{5}+\frac{5976}{84869}a^{4}-\frac{134491}{84869}a^{3}-\frac{1280430}{84869}a^{2}+\frac{4633501}{84869}a+\frac{36326910}{84869}$, $\frac{9330}{84869}a^{5}+\frac{97603}{84869}a^{4}-\frac{1575682}{84869}a^{3}-\frac{12829440}{84869}a^{2}+\frac{68736122}{84869}a+\frac{273912141}{84869}$ | 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: | \( 1692.09459365 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{6}\cdot(2\pi)^{0}\cdot 1692.09459365 \cdot 2}{2\cdot\sqrt{103001481009}}\cr\approx \mathstrut & 0.337429367004 \end{aligned}\]
Galois group
A cyclic group of order 6 |
The 6 conjugacy class representatives for $C_6$ |
Character table for $C_6$ |
Intermediate fields
\(\Q(\sqrt{1281}) \), \(\Q(\zeta_{7})^+\) |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling algebras
Twin sextic algebra: | \(\Q\) $\times$ \(\Q(\sqrt{1281}) \) $\times$ \(\Q(\zeta_{7})^+\) |
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 | ${\href{/padicField/2.3.0.1}{3} }^{2}$ | R | ${\href{/padicField/5.3.0.1}{3} }^{2}$ | R | ${\href{/padicField/11.3.0.1}{3} }^{2}$ | ${\href{/padicField/13.2.0.1}{2} }^{3}$ | ${\href{/padicField/17.6.0.1}{6} }$ | ${\href{/padicField/19.6.0.1}{6} }$ | ${\href{/padicField/23.3.0.1}{3} }^{2}$ | ${\href{/padicField/29.1.0.1}{1} }^{6}$ | ${\href{/padicField/31.3.0.1}{3} }^{2}$ | ${\href{/padicField/37.6.0.1}{6} }$ | ${\href{/padicField/41.1.0.1}{1} }^{6}$ | ${\href{/padicField/43.2.0.1}{2} }^{3}$ | ${\href{/padicField/47.3.0.1}{3} }^{2}$ | ${\href{/padicField/53.3.0.1}{3} }^{2}$ | ${\href{/padicField/59.6.0.1}{6} }$ |
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 |
---|---|---|---|---|---|---|---|
\(3\) | 3.6.3.1 | $x^{6} + 18 x^{2} - 27$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ |
\(7\) | 7.6.5.2 | $x^{6} + 42$ | $6$ | $1$ | $5$ | $C_6$ | $[\ ]_{6}$ |
\(61\) | 61.6.3.2 | $x^{6} + 26047 x^{2} - 13391879$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ |