Normalized defining polynomial
\( x^{12} - 4x^{10} - 2x^{9} + 17x^{8} + 6x^{7} - 30x^{6} - 6x^{5} + 31x^{4} - 26x^{3} - 4x^{2} + 28x - 13 \)
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: | \(-618121839509504\) \(\medspace = -\,2^{18}\cdot 11^{9}\) | 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: | \(17.08\) | 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: | $2^{19/12}11^{3/4}\approx 18.09986519556776$ | ||
Ramified primes: | \(2\), \(11\) | 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: | \(\Q(\sqrt{-11}) \) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{11}a^{8}-\frac{3}{11}a^{7}+\frac{3}{11}a^{6}+\frac{5}{11}a^{5}+\frac{5}{11}a^{4}+\frac{4}{11}a^{3}+\frac{5}{11}a^{2}-\frac{4}{11}a+\frac{4}{11}$, $\frac{1}{55}a^{9}-\frac{1}{55}a^{8}-\frac{14}{55}a^{7}-\frac{1}{5}a^{6}+\frac{26}{55}a^{5}-\frac{8}{55}a^{4}-\frac{9}{55}a^{3}+\frac{17}{55}a^{2}-\frac{3}{11}a+\frac{8}{55}$, $\frac{1}{55}a^{10}-\frac{3}{11}a^{7}+\frac{1}{11}a^{6}-\frac{17}{55}a^{5}+\frac{3}{55}a^{4}+\frac{13}{55}a^{3}+\frac{2}{5}a^{2}-\frac{12}{55}a+\frac{13}{55}$, $\frac{1}{55}a^{11}+\frac{3}{11}a^{7}-\frac{27}{55}a^{6}+\frac{23}{55}a^{5}-\frac{2}{5}a^{4}+\frac{27}{55}a^{3}+\frac{8}{55}a^{2}+\frac{8}{55}a+\frac{1}{11}$
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)
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Fundamental units: | $\frac{51}{55}a^{11}+\frac{41}{55}a^{10}-\frac{173}{55}a^{9}-\frac{22}{5}a^{8}+\frac{62}{5}a^{7}+\frac{861}{55}a^{6}-\frac{872}{55}a^{5}-\frac{206}{11}a^{4}+\frac{817}{55}a^{3}-\frac{626}{55}a^{2}-\frac{779}{55}a+\frac{879}{55}$, $\frac{17}{55}a^{11}+\frac{12}{55}a^{10}-\frac{12}{11}a^{9}-\frac{16}{11}a^{8}+\frac{47}{11}a^{7}+\frac{281}{55}a^{6}-\frac{313}{55}a^{5}-\frac{393}{55}a^{4}+\frac{53}{11}a^{3}-3a^{2}-\frac{18}{5}a+\frac{246}{55}$, $\frac{16}{55}a^{11}+\frac{8}{55}a^{10}-\frac{12}{11}a^{9}-\frac{13}{11}a^{8}+\frac{47}{11}a^{7}+\frac{223}{55}a^{6}-\frac{358}{55}a^{5}-\frac{28}{5}a^{4}+\frac{301}{55}a^{3}-\frac{241}{55}a^{2}-\frac{218}{55}a+\frac{304}{55}$, $\frac{43}{55}a^{11}+\frac{8}{11}a^{10}-\frac{137}{55}a^{9}-\frac{213}{55}a^{8}+\frac{538}{55}a^{7}+\frac{761}{55}a^{6}-\frac{603}{55}a^{5}-\frac{164}{11}a^{4}+\frac{579}{55}a^{3}-\frac{120}{11}a^{2}-\frac{696}{55}a+\frac{659}{55}$, $\frac{87}{55}a^{11}+\frac{67}{55}a^{10}-\frac{301}{55}a^{9}-\frac{404}{55}a^{8}+\frac{1184}{55}a^{7}+\frac{1437}{55}a^{6}-\frac{1579}{55}a^{5}-\frac{346}{11}a^{4}+\frac{1459}{55}a^{3}-\frac{1137}{55}a^{2}-\frac{118}{5}a+\frac{1578}{55}$, $\frac{9}{55}a^{11}+\frac{4}{55}a^{10}-\frac{6}{11}a^{9}-\frac{6}{11}a^{8}+\frac{25}{11}a^{7}+\frac{92}{55}a^{6}-\frac{171}{55}a^{5}-\frac{81}{55}a^{4}+\frac{32}{11}a^{3}-\frac{53}{11}a^{2}-\frac{111}{55}a+\frac{167}{55}$ | 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: | \( 696.143544995 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
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Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{5}\cdot 696.143544995 \cdot 1}{2\cdot\sqrt{618121839509504}}\cr\approx \mathstrut & 0.548391978692 \end{aligned}\]
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
3.1.44.1, 6.2.937024.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 6 siblings: | 6.2.340736.2, 6.0.3748096.1 |
Degree 8 siblings: | 8.4.7256313856.1, 8.0.7256313856.3 |
Degree 12 siblings: | data not computed |
Degree 16 sibling: | data not computed |
Degree 24 siblings: | data not computed |
Minimal sibling: | 6.2.340736.2 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | ${\href{/padicField/3.6.0.1}{6} }^{2}$ | ${\href{/padicField/5.3.0.1}{3} }^{4}$ | ${\href{/padicField/7.4.0.1}{4} }^{3}$ | R | ${\href{/padicField/13.2.0.1}{2} }^{5}{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ | ${\href{/padicField/17.4.0.1}{4} }^{3}$ | ${\href{/padicField/19.2.0.1}{2} }^{5}{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | ${\href{/padicField/23.6.0.1}{6} }^{2}$ | ${\href{/padicField/29.2.0.1}{2} }^{5}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.6.0.1}{6} }^{2}$ | ${\href{/padicField/37.3.0.1}{3} }^{4}$ | ${\href{/padicField/41.2.0.1}{2} }^{5}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.4.0.1}{4} }^{3}$ | ${\href{/padicField/47.2.0.1}{2} }^{4}{,}\,{\href{/padicField/47.1.0.1}{1} }^{4}$ | ${\href{/padicField/53.2.0.1}{2} }^{6}$ | ${\href{/padicField/59.6.0.1}{6} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.12.18.80 | $x^{12} + 2 x^{11} + 2 x^{9} + 2 x^{8} + 2 x^{7} + 2 x^{4} + 2 x^{2} + 2$ | $12$ | $1$ | $18$ | $C_2 \times S_4$ | $[4/3, 4/3, 2]_{3}^{2}$ |
\(11\) | 11.4.3.1 | $x^{4} + 11$ | $4$ | $1$ | $3$ | $D_{4}$ | $[\ ]_{4}^{2}$ |
11.4.3.1 | $x^{4} + 11$ | $4$ | $1$ | $3$ | $D_{4}$ | $[\ ]_{4}^{2}$ | |
11.4.3.2 | $x^{4} + 22$ | $4$ | $1$ | $3$ | $D_{4}$ | $[\ ]_{4}^{2}$ |