Normalized defining polynomial
\( x^{12} - 6x^{11} + 17x^{10} - 28x^{9} + 23x^{8} + 4x^{7} - 34x^{6} + 32x^{5} + 2x^{4} - 16x^{3} + 6x^{2} - 2 \)
Invariants
Degree: | $12$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[2, 5]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(-86029994295296\) \(\medspace = -\,2^{22}\cdot 29^{5}\) | 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: | \(14.50\) | 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: | $2^{9/4}29^{1/2}\approx 25.616305216426433$ | ||
Ramified primes: | \(2\), \(29\) | 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: | \(\Q(\sqrt{-29}) \) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{3}a^{10}+\frac{1}{3}a^{9}-\frac{1}{3}a^{8}+\frac{1}{3}a^{5}+\frac{1}{3}a^{3}+\frac{1}{3}a+\frac{1}{3}$, $\frac{1}{261}a^{11}+\frac{2}{261}a^{10}+\frac{11}{87}a^{9}-\frac{25}{261}a^{8}+\frac{28}{87}a^{7}-\frac{107}{261}a^{6}-\frac{107}{261}a^{5}-\frac{41}{261}a^{4}-\frac{65}{261}a^{3}-\frac{14}{261}a^{2}-\frac{106}{261}a-\frac{65}{261}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$
Unit group
Rank: | $6$ | 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: | $\frac{55}{87}a^{11}-\frac{296}{87}a^{10}+\frac{800}{87}a^{9}-\frac{439}{29}a^{8}+\frac{409}{29}a^{7}-\frac{404}{87}a^{6}-\frac{212}{29}a^{5}+\frac{616}{87}a^{4}+\frac{7}{29}a^{3}+\frac{13}{87}a^{2}-\frac{59}{87}a+\frac{7}{29}$, $a-1$, $\frac{55}{87}a^{11}-\frac{296}{87}a^{10}+\frac{800}{87}a^{9}-\frac{439}{29}a^{8}+\frac{409}{29}a^{7}-\frac{404}{87}a^{6}-\frac{212}{29}a^{5}+\frac{616}{87}a^{4}+\frac{7}{29}a^{3}+\frac{13}{87}a^{2}+\frac{28}{87}a+\frac{7}{29}$, $\frac{179}{261}a^{11}-\frac{1034}{261}a^{10}+\frac{983}{87}a^{9}-\frac{5171}{261}a^{8}+\frac{1793}{87}a^{7}-\frac{2449}{261}a^{6}-\frac{2014}{261}a^{5}+\frac{3362}{261}a^{4}-\frac{1282}{261}a^{3}+\frac{626}{261}a^{2}-\frac{8}{261}a-\frac{499}{261}$, $\frac{97}{87}a^{11}-\frac{647}{87}a^{10}+\frac{2099}{87}a^{9}-\frac{1398}{29}a^{8}+\frac{1730}{29}a^{7}-\frac{3245}{87}a^{6}-\frac{376}{29}a^{5}+\frac{4027}{87}a^{4}-\frac{903}{29}a^{3}+\frac{208}{87}a^{2}+\frac{535}{87}a-\frac{91}{29}$, $\frac{65}{261}a^{11}-\frac{305}{261}a^{10}+\frac{74}{29}a^{9}-\frac{668}{261}a^{8}-\frac{94}{87}a^{7}+\frac{1658}{261}a^{6}-\frac{2170}{261}a^{5}+\frac{467}{261}a^{4}+\frac{1343}{261}a^{3}-\frac{910}{261}a^{2}-\frac{17}{261}a+\frac{299}{261}$ | 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: | \( 238.080953447 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{5}\cdot 238.080953447 \cdot 1}{2\cdot\sqrt{86029994295296}}\cr\approx \mathstrut & 0.502723333933 \end{aligned}\]
Galois group
$C_2\times S_4$ (as 12T22):
A solvable group of order 48 |
The 10 conjugacy class representatives for $C_2 \times S_4$ |
Character table for $C_2 \times S_4$ |
Intermediate fields
3.1.116.1, 6.2.215296.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 6 siblings: | 6.0.430592.2, 6.2.12487168.1 |
Degree 8 siblings: | 8.0.55115776.2, 8.4.46352367616.2 |
Degree 12 siblings: | data not computed |
Degree 16 sibling: | data not computed |
Degree 24 siblings: | data not computed |
Minimal sibling: | 6.0.430592.2 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | ${\href{/padicField/3.3.0.1}{3} }^{4}$ | ${\href{/padicField/5.6.0.1}{6} }^{2}$ | ${\href{/padicField/7.4.0.1}{4} }^{3}$ | ${\href{/padicField/11.3.0.1}{3} }^{4}$ | ${\href{/padicField/13.6.0.1}{6} }^{2}$ | ${\href{/padicField/17.4.0.1}{4} }^{3}$ | ${\href{/padicField/19.2.0.1}{2} }^{6}$ | ${\href{/padicField/23.4.0.1}{4} }^{3}$ | R | ${\href{/padicField/31.6.0.1}{6} }^{2}$ | ${\href{/padicField/37.2.0.1}{2} }^{5}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.4.0.1}{4} }^{3}$ | ${\href{/padicField/43.3.0.1}{3} }^{4}$ | ${\href{/padicField/47.6.0.1}{6} }^{2}$ | ${\href{/padicField/53.6.0.1}{6} }^{2}$ | ${\href{/padicField/59.4.0.1}{4} }^{3}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.4.4.4 | $x^{4} - 2 x^{3} + 4 x^{2} + 12 x + 12$ | $2$ | $2$ | $4$ | $D_{4}$ | $[2, 2]^{2}$ |
2.8.18.56 | $x^{8} + 4 x^{7} + 2 x^{6} + 4 x^{3} + 14$ | $8$ | $1$ | $18$ | $D_4\times C_2$ | $[2, 2, 3]^{2}$ | |
\(29\) | $\Q_{29}$ | $x + 27$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{29}$ | $x + 27$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
29.2.1.1 | $x^{2} + 29$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
29.4.2.1 | $x^{4} + 1440 x^{3} + 535166 x^{2} + 12071520 x + 1504089$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
29.4.2.1 | $x^{4} + 1440 x^{3} + 535166 x^{2} + 12071520 x + 1504089$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |