Normalized defining polynomial
\( x^{12} - 8x^{9} + 6x^{7} + 34x^{6} + 16x^{5} + 6x^{4} - 20x^{3} - 10x^{2} - 8x + 2 \)
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: | \(-38537381281792\) \(\medspace = -\,2^{18}\cdot 43^{5}\) | 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: | \(13.56\) | 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: | $2^{19/12}43^{1/2}\approx 19.65011309429094$ | ||
Ramified primes: | \(2\), \(43\) | 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: | \(\Q(\sqrt{-43}) \) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{3036017}a^{11}+\frac{1347081}{3036017}a^{10}-\frac{140339}{3036017}a^{9}-\frac{1293911}{3036017}a^{8}-\frac{1275955}{3036017}a^{7}-\frac{1036952}{3036017}a^{6}+\frac{940554}{3036017}a^{5}-\frac{335618}{3036017}a^{4}+\frac{804486}{3036017}a^{3}-\frac{1498821}{3036017}a^{2}-\frac{14052}{3036017}a+\frac{383775}{3036017}$
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: | $\frac{24140}{3036017}a^{11}-\frac{242747}{3036017}a^{10}+\frac{411512}{3036017}a^{9}-\frac{468644}{3036017}a^{8}+\frac{1874782}{3036017}a^{7}-\frac{3097132}{3036017}a^{6}+\frac{1638434}{3036017}a^{5}-\frac{4761181}{3036017}a^{4}+\frac{7999342}{3036017}a^{3}-\frac{4360368}{3036017}a^{2}+\frac{818624}{3036017}a-\frac{4631401}{3036017}$, $\frac{658891}{3036017}a^{11}-\frac{22779}{3036017}a^{10}-\frac{134280}{3036017}a^{9}-\frac{5414948}{3036017}a^{8}+\frac{345633}{3036017}a^{7}+\frac{5141550}{3036017}a^{6}+\frac{22919642}{3036017}a^{5}+\frac{7798642}{3036017}a^{4}-\frac{1767072}{3036017}a^{3}-\frac{16201819}{3036017}a^{2}-\frac{4956516}{3036017}a-\frac{3962405}{3036017}$, $\frac{627327}{3036017}a^{11}+\frac{130622}{3036017}a^{10}-\frac{22887}{3036017}a^{9}-\frac{4908828}{3036017}a^{8}-\frac{1212269}{3036017}a^{7}+\frac{4195201}{3036017}a^{6}+\frac{21447412}{3036017}a^{5}+\frac{14653915}{3036017}a^{4}+\frac{4755046}{3036017}a^{3}-\frac{11596652}{3036017}a^{2}-\frac{7713687}{3036017}a-\frac{3728675}{3036017}$, $\frac{261868}{3036017}a^{11}-\frac{443939}{3036017}a^{10}+\frac{692533}{3036017}a^{9}-\frac{2244480}{3036017}a^{8}+\frac{3139029}{3036017}a^{7}-\frac{3185856}{3036017}a^{6}+\frac{7151764}{3036017}a^{5}-\frac{4030325}{3036017}a^{4}+\frac{12064286}{3036017}a^{3}-\frac{3051902}{3036017}a^{2}+\frac{2919485}{3036017}a-\frac{2879051}{3036017}$, $\frac{627327}{3036017}a^{11}+\frac{130622}{3036017}a^{10}-\frac{22887}{3036017}a^{9}-\frac{4908828}{3036017}a^{8}-\frac{1212269}{3036017}a^{7}+\frac{4195201}{3036017}a^{6}+\frac{21447412}{3036017}a^{5}+\frac{14653915}{3036017}a^{4}+\frac{4755046}{3036017}a^{3}-\frac{8560635}{3036017}a^{2}-\frac{7713687}{3036017}a-\frac{3728675}{3036017}$, $\frac{116865}{3036017}a^{11}+\frac{31564}{3036017}a^{10}-\frac{153401}{3036017}a^{9}-\frac{1046313}{3036017}a^{8}-\frac{506120}{3036017}a^{7}+\frac{2259092}{3036017}a^{6}+\frac{4919759}{3036017}a^{5}+\frac{3342070}{3036017}a^{4}-\frac{6154083}{3036017}a^{3}-\frac{8859418}{3036017}a^{2}-\frac{5773817}{3036017}a+\frac{1822251}{3036017}$ | sage: UK.fundamental_units()
gp: K.fu
magma: [K|fUK(g): g in Generators(UK)];
oscar: [K(fUK(a)) for a in gens(UK)]
| |
Regulator: | \( 122.014624052 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{5}\cdot 122.014624052 \cdot 1}{2\cdot\sqrt{38537381281792}}\cr\approx \mathstrut & 0.384946276943 \end{aligned}\]
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
3.1.172.1, 6.2.118336.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 6 siblings: | 6.0.473344.2, 6.2.20353792.3 |
Degree 8 siblings: | 8.0.7573504.1, 8.4.14003408896.2 |
Degree 12 siblings: | data not computed |
Degree 16 sibling: | data not computed |
Degree 24 siblings: | data not computed |
Minimal sibling: | 6.0.473344.2 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | ${\href{/padicField/3.4.0.1}{4} }^{3}$ | ${\href{/padicField/5.4.0.1}{4} }^{3}$ | ${\href{/padicField/7.4.0.1}{4} }^{3}$ | ${\href{/padicField/11.6.0.1}{6} }^{2}$ | ${\href{/padicField/13.3.0.1}{3} }^{4}$ | ${\href{/padicField/17.3.0.1}{3} }^{4}$ | ${\href{/padicField/19.2.0.1}{2} }^{5}{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | ${\href{/padicField/23.6.0.1}{6} }^{2}$ | ${\href{/padicField/29.4.0.1}{4} }^{3}$ | ${\href{/padicField/31.6.0.1}{6} }^{2}$ | ${\href{/padicField/37.2.0.1}{2} }^{5}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.3.0.1}{3} }^{4}$ | R | ${\href{/padicField/47.2.0.1}{2} }^{4}{,}\,{\href{/padicField/47.1.0.1}{1} }^{4}$ | ${\href{/padicField/53.3.0.1}{3} }^{4}$ | ${\href{/padicField/59.2.0.1}{2} }^{4}{,}\,{\href{/padicField/59.1.0.1}{1} }^{4}$ |
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 | $x^{12} + 2 x^{11} + 2 x^{9} + 2 x^{8} + 2 x^{7} + 2 x^{4} + 2 x^{2} + 2$ | $12$ | $1$ | $18$ | $C_2 \times S_4$ | $[4/3, 4/3, 2]_{3}^{2}$ |
\(43\) | 43.2.0.1 | $x^{2} + 42 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
43.2.1.1 | $x^{2} + 86$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
43.4.2.1 | $x^{4} + 84 x^{3} + 1856 x^{2} + 3864 x + 77452$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
43.4.2.1 | $x^{4} + 84 x^{3} + 1856 x^{2} + 3864 x + 77452$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |