Normalized defining polynomial
\( x^{12} - 6x^{10} - 10x^{9} - 9x^{8} + 16x^{6} + 6x^{5} - 6x^{3} - 6x^{2} + 6x - 1 \)
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: | \(11284439629824\) \(\medspace = 2^{18}\cdot 3^{16}\) | 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: | \(12.24\) | 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: | $2^{7/4}3^{4/3}\approx 14.553389922872519$ | ||
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: | \(\Q\) | ||
$\card{ \Aut(K/\Q) }$: | $4$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{3}a^{9}+\frac{1}{3}a^{3}+\frac{1}{3}$, $\frac{1}{3}a^{10}+\frac{1}{3}a^{4}+\frac{1}{3}a$, $\frac{1}{1563}a^{11}-\frac{6}{521}a^{10}-\frac{203}{1563}a^{9}-\frac{1}{521}a^{8}+\frac{15}{521}a^{7}+\frac{251}{521}a^{6}+\frac{529}{1563}a^{5}-\frac{46}{521}a^{4}+\frac{400}{1563}a^{3}+\frac{88}{1563}a^{2}-\frac{9}{521}a-\frac{29}{1563}$
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: | $\frac{2080}{1563}a^{11}+\frac{545}{521}a^{10}-\frac{11171}{1563}a^{9}-\frac{9895}{521}a^{8}-\frac{14127}{521}a^{7}-\frac{10903}{521}a^{6}+\frac{9346}{1563}a^{5}+\frac{6957}{521}a^{4}+\frac{17677}{1563}a^{3}+\frac{1732}{1563}a^{2}-\frac{4132}{521}a+\frac{3763}{1563}$, $\frac{125}{1563}a^{11}+\frac{355}{1563}a^{10}-\frac{296}{521}a^{9}-\frac{1167}{521}a^{8}-\frac{1251}{521}a^{7}-\frac{406}{521}a^{6}+\frac{3605}{1563}a^{5}+\frac{7237}{1563}a^{4}+\frac{863}{521}a^{3}+\frac{59}{1563}a^{2}-\frac{770}{1563}a-\frac{340}{521}$, $\frac{1779}{521}a^{11}+\frac{1322}{521}a^{10}-\frac{29428}{1563}a^{9}-\frac{25135}{521}a^{8}-\frac{34044}{521}a^{7}-\frac{23870}{521}a^{6}+\frac{12669}{521}a^{5}+\frac{21771}{521}a^{4}+\frac{47153}{1563}a^{3}+\frac{252}{521}a^{2}-\frac{11563}{521}a+\frac{5174}{1563}$, $\frac{115}{1563}a^{11}-\frac{169}{521}a^{10}-\frac{1463}{1563}a^{9}+\frac{406}{521}a^{8}+\frac{2767}{521}a^{7}+\frac{5420}{521}a^{6}+\frac{17071}{1563}a^{5}+\frac{962}{521}a^{4}-\frac{7142}{1563}a^{3}-\frac{8636}{1563}a^{2}-\frac{1035}{521}a+\frac{4480}{1563}$, $\frac{695}{521}a^{11}+\frac{1024}{1563}a^{10}-\frac{4062}{521}a^{9}-\frac{8858}{521}a^{8}-\frac{10405}{521}a^{7}-\frac{4959}{521}a^{6}+\frac{9207}{521}a^{5}+\frac{27475}{1563}a^{4}+\frac{4475}{521}a^{3}-\frac{1360}{521}a^{2}-\frac{17741}{1563}a+\frac{1727}{521}$, $\frac{340}{521}a^{11}-\frac{125}{1563}a^{10}-\frac{6475}{1563}a^{9}-\frac{3104}{521}a^{8}-\frac{1893}{521}a^{7}+\frac{1251}{521}a^{6}+\frac{5846}{521}a^{5}+\frac{2515}{1563}a^{4}-\frac{7237}{1563}a^{3}-\frac{2903}{521}a^{2}-\frac{6179}{1563}a+\frac{6890}{1563}$, $\frac{1537}{1563}a^{11}+\frac{1510}{1563}a^{10}-\frac{2756}{521}a^{9}-\frac{7789}{521}a^{8}-\frac{11331}{521}a^{7}-\frac{9652}{521}a^{6}+\frac{1876}{1563}a^{5}+\frac{14008}{1563}a^{4}+\frac{3480}{521}a^{3}+\frac{2401}{1563}a^{2}-\frac{9197}{1563}a+\frac{946}{521}$ | 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: | \( 63.1219944158 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
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Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{4}\cdot 63.1219944158 \cdot 1}{2\cdot\sqrt{11284439629824}}\cr\approx \mathstrut & 0.234288069665 \end{aligned}\]
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
\(\Q(\zeta_{9})^+\), 6.4.419904.1 x2, 6.2.419904.1 |
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: | 6.4.419904.1 |
Degree 8 sibling: | 8.0.107495424.1 |
Degree 12 sibling: | 12.0.11284439629824.2 |
Minimal sibling: | 6.4.419904.1 |
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 | ${\href{/padicField/5.3.0.1}{3} }^{4}$ | ${\href{/padicField/7.6.0.1}{6} }^{2}$ | ${\href{/padicField/11.6.0.1}{6} }^{2}$ | ${\href{/padicField/13.3.0.1}{3} }^{4}$ | ${\href{/padicField/17.2.0.1}{2} }^{6}$ | ${\href{/padicField/19.2.0.1}{2} }^{4}{,}\,{\href{/padicField/19.1.0.1}{1} }^{4}$ | ${\href{/padicField/23.6.0.1}{6} }^{2}$ | ${\href{/padicField/29.3.0.1}{3} }^{4}$ | ${\href{/padicField/31.6.0.1}{6} }^{2}$ | ${\href{/padicField/37.2.0.1}{2} }^{6}$ | ${\href{/padicField/41.3.0.1}{3} }^{4}$ | ${\href{/padicField/43.6.0.1}{6} }^{2}$ | ${\href{/padicField/47.6.0.1}{6} }^{2}$ | ${\href{/padicField/53.2.0.1}{2} }^{6}$ | ${\href{/padicField/59.6.0.1}{6} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.12.18.31 | $x^{12} - 4 x^{11} + 22 x^{10} + 16 x^{9} + 50 x^{8} + 32 x^{7} + 144 x^{6} + 96 x^{5} + 236 x^{4} + 248 x^{2} + 248$ | $4$ | $3$ | $18$ | $A_4\times C_2$ | $[2, 2, 2]^{3}$ |
\(3\) | 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 | $x^{6} + 18 x^{5} + 114 x^{4} + 326 x^{3} + 570 x^{2} + 528 x + 197$ | $3$ | $2$ | $8$ | $C_6$ | $[2]^{2}$ |