Normalized defining polynomial
\( x^{12} - 6x^{11} + 55x^{9} - 84x^{8} - 60x^{7} + 189x^{6} - 60x^{5} - 84x^{4} + 55x^{3} - 6x + 1 \)
Invariants
Degree: | $12$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[12, 0]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(136772361580078125\) \(\medspace = 3^{14}\cdot 5^{9}\cdot 11^{4}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
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Root discriminant: | \(26.79\) | 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))
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Galois root discriminant: | $3^{25/18}5^{3/4}11^{2/3}\approx 76.06024196897057$ | ||
Ramified primes: | \(3\), \(5\), \(11\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
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Discriminant root field: | \(\Q(\sqrt{5}) \) | ||
$\card{ \Aut(K/\Q) }$: | $6$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
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This field is not Galois over $\Q$. | |||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3}a^{6}-\frac{1}{3}a^{3}+\frac{1}{3}$, $\frac{1}{3}a^{7}-\frac{1}{3}a^{4}+\frac{1}{3}a$, $\frac{1}{3}a^{8}-\frac{1}{3}a^{5}+\frac{1}{3}a^{2}$, $\frac{1}{3}a^{9}+\frac{1}{3}$, $\frac{1}{3}a^{10}+\frac{1}{3}a$, $\frac{1}{3}a^{11}+\frac{1}{3}a^{2}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
Rank: | $11$ | 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)
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Fundamental units: | $a^{10}-5a^{9}-\frac{16}{3}a^{8}+\frac{154}{3}a^{7}-31a^{6}-\frac{323}{3}a^{5}+\frac{278}{3}a^{4}+64a^{3}-\frac{154}{3}a^{2}-\frac{26}{3}a+5$, $9a^{11}-49a^{10}-\frac{82}{3}a^{9}+\frac{1441}{3}a^{8}-\frac{1465}{3}a^{7}-817a^{6}+\frac{3746}{3}a^{5}+\frac{505}{3}a^{4}-672a^{3}+\frac{334}{3}a^{2}+\frac{212}{3}a-\frac{46}{3}$, $11a^{11}-60a^{10}-\frac{97}{3}a^{9}+\frac{1756}{3}a^{8}-\frac{1819}{3}a^{7}-971a^{6}+\frac{4598}{3}a^{5}+\frac{430}{3}a^{4}-805a^{3}+\frac{517}{3}a^{2}+\frac{233}{3}a-\frac{61}{3}$, $a$, $a-1$, $\frac{19}{3}a^{11}-\frac{94}{3}a^{10}-\frac{103}{3}a^{9}+319a^{8}-\frac{569}{3}a^{7}-\frac{1942}{3}a^{6}+562a^{5}+\frac{1013}{3}a^{4}-\frac{884}{3}a^{3}-\frac{62}{3}a^{2}+32a-\frac{8}{3}$, $\frac{1}{3}a^{11}-\frac{2}{3}a^{10}-\frac{20}{3}a^{9}+\frac{35}{3}a^{8}+39a^{7}-66a^{6}-\frac{200}{3}a^{5}+112a^{4}+31a^{3}-55a^{2}-\frac{5}{3}a+\frac{19}{3}$, $\frac{5}{3}a^{11}-\frac{32}{3}a^{10}+\frac{8}{3}a^{9}+\frac{293}{3}a^{8}-167a^{7}-\frac{319}{3}a^{6}+\frac{1114}{3}a^{5}-89a^{4}-\frac{536}{3}a^{3}+\frac{202}{3}a^{2}+\frac{58}{3}a-\frac{20}{3}$, $5a^{11}-\frac{82}{3}a^{10}-15a^{9}+\frac{805}{3}a^{8}-272a^{7}-\frac{1376}{3}a^{6}+\frac{2063}{3}a^{5}+98a^{4}-\frac{1072}{3}a^{3}+\frac{181}{3}a^{2}+\frac{95}{3}a-\frac{20}{3}$, $\frac{4}{3}a^{11}-\frac{19}{3}a^{10}-\frac{26}{3}a^{9}+\frac{200}{3}a^{8}-27a^{7}-153a^{6}+\frac{316}{3}a^{5}+107a^{4}-73a^{3}-20a^{2}+\frac{29}{3}a+\frac{4}{3}$, $\frac{41}{3}a^{11}-76a^{10}-\frac{98}{3}a^{9}+\frac{2201}{3}a^{8}-\frac{2485}{3}a^{7}-1149a^{6}+\frac{6136}{3}a^{5}+\frac{82}{3}a^{4}-1062a^{3}+\frac{862}{3}a^{2}+\frac{311}{3}a-\frac{107}{3}$ (assuming GRH) | sage: UK.fundamental_units()
gp: K.fu
magma: [K|fUK(g): g in Generators(UK)];
oscar: [K(fUK(a)) for a in gens(UK)]
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Regulator: | \( 59761.38389 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
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Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{12}\cdot(2\pi)^{0}\cdot 59761.38389 \cdot 1}{2\cdot\sqrt{136772361580078125}}\cr\approx \mathstrut & 0.3309416480 \end{aligned}\] (assuming GRH)
Galois group
$C_3^2:C_4$ (as 12T17):
A solvable group of order 36 |
The 6 conjugacy class representatives for $(C_3\times C_3):C_4$ |
Character table for $(C_3\times C_3):C_4$ |
Intermediate fields
\(\Q(\sqrt{5}) \), \(\Q(\zeta_{15})^+\), 6.6.55130625.1 x3 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Galois closure: | deg 36 |
Degree 6 siblings: | 6.6.6670805625.2, 6.6.55130625.1 |
Degree 9 sibling: | 9.9.14710627334390625.1 |
Degree 12 sibling: | deg 12 |
Degree 18 sibling: | deg 18 |
Minimal sibling: | 6.6.55130625.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | ${\href{/padicField/2.4.0.1}{4} }^{3}$ | R | R | ${\href{/padicField/7.4.0.1}{4} }^{3}$ | R | ${\href{/padicField/13.4.0.1}{4} }^{3}$ | ${\href{/padicField/17.4.0.1}{4} }^{3}$ | ${\href{/padicField/19.2.0.1}{2} }^{6}$ | ${\href{/padicField/23.4.0.1}{4} }^{3}$ | ${\href{/padicField/29.3.0.1}{3} }^{4}$ | ${\href{/padicField/31.3.0.1}{3} }^{4}$ | ${\href{/padicField/37.4.0.1}{4} }^{3}$ | ${\href{/padicField/41.2.0.1}{2} }^{6}$ | ${\href{/padicField/43.4.0.1}{4} }^{3}$ | ${\href{/padicField/47.4.0.1}{4} }^{3}$ | ${\href{/padicField/53.4.0.1}{4} }^{3}$ | ${\href{/padicField/59.3.0.1}{3} }^{4}$ |
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 |
---|---|---|---|---|---|---|---|
\(3\) | 3.12.14.14 | $x^{12} - 6 x^{8} - 6 x^{6} + 45 x^{4} + 54 x^{2} + 18$ | $6$ | $2$ | $14$ | $(C_3\times C_3):C_4$ | $[3/2, 3/2]_{2}^{2}$ |
\(5\) | 5.4.3.1 | $x^{4} + 20$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ |
5.4.3.1 | $x^{4} + 20$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ | |
5.4.3.1 | $x^{4} + 20$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ | |
\(11\) | 11.2.0.1 | $x^{2} + 7 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
11.2.0.1 | $x^{2} + 7 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
11.2.0.1 | $x^{2} + 7 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
11.6.4.1 | $x^{6} + 21 x^{5} + 153 x^{4} + 449 x^{3} + 537 x^{2} + 1569 x + 3440$ | $3$ | $2$ | $4$ | $S_3$ | $[\ ]_{3}^{2}$ |