Normalized defining polynomial
\( x^{12} - 6 x^{11} + 18 x^{10} - 28 x^{9} + 12 x^{8} + 36 x^{7} - 67 x^{6} + 36 x^{5} + 12 x^{4} + \cdots + 1 \)
Invariants
Degree: | $12$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[4, 4]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(5609891727441\) \(\medspace = 3^{16}\cdot 19^{4}\) | 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: | \(11.55\) | 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: | $3^{4/3}19^{1/2}\approx 18.859860385004858$ | ||
Ramified primes: | \(3\), \(19\) | 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: | \(\Q\) | ||
$\card{ \Aut(K/\Q) }$: | $4$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{3}-\frac{1}{2}$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{4}-\frac{1}{2}a$, $\frac{1}{2}a^{8}-\frac{1}{2}a^{5}-\frac{1}{2}a^{2}$, $\frac{1}{6}a^{9}+\frac{1}{6}a^{6}+\frac{1}{6}a^{3}-\frac{1}{3}$, $\frac{1}{6}a^{10}+\frac{1}{6}a^{7}+\frac{1}{6}a^{4}-\frac{1}{3}a$, $\frac{1}{6}a^{11}+\frac{1}{6}a^{8}+\frac{1}{6}a^{5}-\frac{1}{3}a^{2}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$
Unit group
Rank: | $7$ | 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: | $\frac{1}{6}a^{11}-\frac{1}{3}a^{10}-\frac{5}{6}a^{9}+\frac{37}{6}a^{8}-\frac{77}{6}a^{7}+\frac{43}{6}a^{6}+\frac{103}{6}a^{5}-\frac{191}{6}a^{4}+\frac{55}{6}a^{3}+\frac{41}{3}a^{2}-\frac{71}{6}a+\frac{11}{3}$, $a$, $\frac{11}{3}a^{11}-\frac{39}{2}a^{10}+\frac{313}{6}a^{9}-\frac{193}{3}a^{8}-7a^{7}+\frac{811}{6}a^{6}-\frac{451}{3}a^{5}+9a^{4}+\frac{415}{6}a^{3}-\frac{157}{3}a^{2}+\frac{37}{2}a-\frac{7}{3}$, $\frac{11}{3}a^{11}-\frac{39}{2}a^{10}+\frac{313}{6}a^{9}-\frac{193}{3}a^{8}-7a^{7}+\frac{811}{6}a^{6}-\frac{451}{3}a^{5}+9a^{4}+\frac{415}{6}a^{3}-\frac{157}{3}a^{2}+\frac{39}{2}a-\frac{10}{3}$, $\frac{3}{2}a^{11}-\frac{26}{3}a^{10}+\frac{74}{3}a^{9}-\frac{69}{2}a^{8}+\frac{16}{3}a^{7}+\frac{361}{6}a^{6}-\frac{167}{2}a^{5}+\frac{61}{3}a^{4}+\frac{205}{6}a^{3}-29a^{2}+\frac{37}{3}a-\frac{17}{6}$, $\frac{3}{2}a^{11}-\frac{28}{3}a^{10}+\frac{57}{2}a^{9}-45a^{8}+\frac{56}{3}a^{7}+\frac{125}{2}a^{6}-115a^{5}+\frac{167}{3}a^{4}+\frac{71}{2}a^{3}-\frac{105}{2}a^{2}+\frac{71}{3}a-5$, $a-1$ | 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: | \( 38.5547536132 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
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Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{4}\cdot 38.5547536132 \cdot 1}{2\cdot\sqrt{5609891727441}}\cr\approx \mathstrut & 0.202959859159 \end{aligned}\]
Galois group
$C_2\times A_4$ (as 12T6):
A solvable group of order 24 |
The 8 conjugacy class representatives for $A_4\times C_2$ |
Character table for $A_4\times C_2$ |
Intermediate fields
\(\Q(\zeta_{9})^+\), 6.4.124659.1 x2, 6.2.2368521.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Galois closure: | deg 24 |
Degree 6 sibling: | 6.4.124659.1 |
Degree 8 sibling: | 8.0.855036081.1 |
Degree 12 sibling: | 12.0.2025170913606201.1 |
Minimal sibling: | 6.4.124659.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | ${\href{/padicField/2.6.0.1}{6} }^{2}$ | R | ${\href{/padicField/5.3.0.1}{3} }^{4}$ | ${\href{/padicField/7.3.0.1}{3} }^{4}$ | ${\href{/padicField/11.3.0.1}{3} }^{4}$ | ${\href{/padicField/13.6.0.1}{6} }^{2}$ | ${\href{/padicField/17.2.0.1}{2} }^{6}$ | R | ${\href{/padicField/23.3.0.1}{3} }^{4}$ | ${\href{/padicField/29.6.0.1}{6} }^{2}$ | ${\href{/padicField/31.6.0.1}{6} }^{2}$ | ${\href{/padicField/37.2.0.1}{2} }^{4}{,}\,{\href{/padicField/37.1.0.1}{1} }^{4}$ | ${\href{/padicField/41.6.0.1}{6} }^{2}$ | ${\href{/padicField/43.3.0.1}{3} }^{4}$ | ${\href{/padicField/47.3.0.1}{3} }^{4}$ | ${\href{/padicField/53.2.0.1}{2} }^{4}{,}\,{\href{/padicField/53.1.0.1}{1} }^{4}$ | ${\href{/padicField/59.6.0.1}{6} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(3\) | 3.6.8.3 | $x^{6} + 18 x^{5} + 114 x^{4} + 326 x^{3} + 570 x^{2} + 528 x + 197$ | $3$ | $2$ | $8$ | $C_6$ | $[2]^{2}$ |
3.6.8.3 | $x^{6} + 18 x^{5} + 114 x^{4} + 326 x^{3} + 570 x^{2} + 528 x + 197$ | $3$ | $2$ | $8$ | $C_6$ | $[2]^{2}$ | |
\(19\) | 19.2.0.1 | $x^{2} + 18 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
19.2.0.1 | $x^{2} + 18 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
19.4.2.1 | $x^{4} + 36 x^{3} + 366 x^{2} + 756 x + 6445$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
19.4.2.1 | $x^{4} + 36 x^{3} + 366 x^{2} + 756 x + 6445$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |