Normalized defining polynomial
\( x^{12} - 9x^{10} + 33x^{8} - 54x^{6} + 45x^{4} - 18x^{2} + 3 \)
Invariants
Degree: | $12$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[0, 6]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(4760622968832\) \(\medspace = 2^{12}\cdot 3^{19}\) | 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: | \(11.39\) | 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: | $2^{3/2}3^{19/12}\approx 16.105973497991872$ | ||
Ramified primes: | \(2\), \(3\) | 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: | \(\Q(\sqrt{3}) \) | ||
$\card{ \Aut(K/\Q) }$: | $6$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{37}a^{10}-\frac{13}{37}a^{8}+\frac{11}{37}a^{6}+\frac{13}{37}a^{4}-\frac{7}{37}a^{2}+\frac{10}{37}$, $\frac{1}{37}a^{11}-\frac{13}{37}a^{9}+\frac{11}{37}a^{7}+\frac{13}{37}a^{5}-\frac{7}{37}a^{3}+\frac{10}{37}a$
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)
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Torsion generator: | \( \frac{18}{37} a^{10} - \frac{160}{37} a^{8} + \frac{568}{37} a^{6} - \frac{839}{37} a^{4} + \frac{503}{37} a^{2} - \frac{79}{37} \) (order $18$) | 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: | $\frac{11}{37}a^{10}-\frac{106}{37}a^{8}+\frac{417}{37}a^{6}-\frac{745}{37}a^{4}+\frac{589}{37}a^{2}-\frac{112}{37}$, $\frac{13}{37}a^{10}-\frac{95}{37}a^{8}+\frac{254}{37}a^{6}-\frac{164}{37}a^{4}-\frac{17}{37}a^{2}+\frac{19}{37}$, $\frac{11}{37}a^{11}-\frac{13}{37}a^{10}-\frac{106}{37}a^{9}+\frac{95}{37}a^{8}+\frac{417}{37}a^{7}-\frac{254}{37}a^{6}-\frac{745}{37}a^{5}+\frac{164}{37}a^{4}+\frac{589}{37}a^{3}+\frac{17}{37}a^{2}-\frac{149}{37}a-\frac{19}{37}$, $\frac{7}{37}a^{11}+\frac{7}{37}a^{10}-\frac{54}{37}a^{9}-\frac{54}{37}a^{8}+\frac{151}{37}a^{7}+\frac{151}{37}a^{6}-\frac{94}{37}a^{5}-\frac{94}{37}a^{4}-\frac{86}{37}a^{3}-\frac{86}{37}a^{2}+\frac{70}{37}a+\frac{70}{37}$, $\frac{45}{37}a^{11}-\frac{24}{37}a^{10}-\frac{400}{37}a^{9}+\frac{201}{37}a^{8}+\frac{1420}{37}a^{7}-\frac{671}{37}a^{6}-\frac{2116}{37}a^{5}+\frac{909}{37}a^{4}+\frac{1350}{37}a^{3}-\frac{609}{37}a^{2}-\frac{290}{37}a+\frac{130}{37}$ | 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: | \( 216.234636806 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
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Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 216.234636806 \cdot 1}{18\cdot\sqrt{4760622968832}}\cr\approx \mathstrut & 0.338766199838 \end{aligned}\]
Galois group
$C_3\times D_4$ (as 12T14):
A solvable group of order 24 |
The 15 conjugacy class representatives for $D_4 \times C_3$ |
Character table for $D_4 \times C_3$ |
Intermediate fields
\(\Q(\sqrt{-3}) \), \(\Q(\zeta_{9})^+\), 4.0.432.1, \(\Q(\zeta_{9})\) |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Galois closure: | deg 24 |
Degree 12 sibling: | 12.6.304679870005248.2 |
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 | R | R | ${\href{/padicField/5.12.0.1}{12} }$ | ${\href{/padicField/7.6.0.1}{6} }{,}\,{\href{/padicField/7.3.0.1}{3} }^{2}$ | ${\href{/padicField/11.6.0.1}{6} }^{2}$ | ${\href{/padicField/13.6.0.1}{6} }^{2}$ | ${\href{/padicField/17.4.0.1}{4} }^{3}$ | ${\href{/padicField/19.2.0.1}{2} }^{3}{,}\,{\href{/padicField/19.1.0.1}{1} }^{6}$ | ${\href{/padicField/23.6.0.1}{6} }^{2}$ | ${\href{/padicField/29.12.0.1}{12} }$ | ${\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.3.0.1}{3} }^{2}$ | ${\href{/padicField/37.1.0.1}{1} }^{12}$ | ${\href{/padicField/41.12.0.1}{12} }$ | ${\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.3.0.1}{3} }^{2}$ | ${\href{/padicField/47.6.0.1}{6} }^{2}$ | ${\href{/padicField/53.4.0.1}{4} }^{3}$ | ${\href{/padicField/59.6.0.1}{6} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.12.12.18 | $x^{12} - 10 x^{11} + 72 x^{10} - 332 x^{9} + 1316 x^{8} - 4160 x^{7} + 12128 x^{6} - 27904 x^{5} + 53744 x^{4} - 69600 x^{3} + 71680 x^{2} - 41536 x + 38848$ | $2$ | $6$ | $12$ | $D_4 \times C_3$ | $[2, 2]^{6}$ |
\(3\) | 3.12.19.43 | $x^{12} + 6 x^{8} + 3$ | $12$ | $1$ | $19$ | $D_4 \times C_3$ | $[2]_{4}^{2}$ |