Normalized defining polynomial
\( x^{12} - 8x^{10} - 4x^{9} + 11x^{8} - 16x^{7} - 28x^{6} + 80x^{5} + 182x^{4} + 144x^{3} + 58x^{2} + 12x + 1 \)
Invariants
Degree: | $12$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[6, 3]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(-677021181018112\) \(\medspace = -\,2^{24}\cdot 7^{9}\) | 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: | \(17.21\) | 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: | $2^{2}7^{5/6}\approx 20.244560739185545$ | ||
Ramified primes: | \(2\), \(7\) | 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: | \(\Q(\sqrt{-7}) \) | ||
$\card{ \Aut(K/\Q) }$: | $6$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{7}a^{9}-\frac{2}{7}a^{8}-\frac{3}{7}a^{7}+\frac{1}{7}a^{6}-\frac{3}{7}a^{5}+\frac{2}{7}a^{4}+\frac{1}{7}a^{3}+\frac{3}{7}a-\frac{1}{7}$, $\frac{1}{7}a^{10}+\frac{2}{7}a^{7}-\frac{1}{7}a^{6}+\frac{3}{7}a^{5}-\frac{2}{7}a^{4}+\frac{2}{7}a^{3}+\frac{3}{7}a^{2}-\frac{2}{7}a-\frac{2}{7}$, $\frac{1}{49}a^{11}-\frac{3}{49}a^{10}+\frac{1}{49}a^{9}-\frac{1}{7}a^{8}-\frac{17}{49}a^{7}-\frac{2}{7}a^{6}+\frac{2}{7}a^{5}-\frac{11}{49}a^{4}+\frac{19}{49}a^{3}-\frac{11}{49}a^{2}-\frac{1}{7}a-\frac{16}{49}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$
Unit group
Rank: | $8$ | 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: | $\frac{176}{49}a^{11}+\frac{4}{49}a^{10}-\frac{1448}{49}a^{9}-\frac{97}{7}a^{8}+\frac{2160}{49}a^{7}-\frac{422}{7}a^{6}-\frac{740}{7}a^{5}+\frac{14871}{49}a^{4}+\frac{32184}{49}a^{3}+\frac{23082}{49}a^{2}+\frac{1104}{7}a+\frac{1027}{49}$, $\frac{176}{49}a^{11}+\frac{4}{49}a^{10}-\frac{1448}{49}a^{9}-\frac{97}{7}a^{8}+\frac{2160}{49}a^{7}-\frac{422}{7}a^{6}-\frac{740}{7}a^{5}+\frac{14871}{49}a^{4}+\frac{32184}{49}a^{3}+\frac{23082}{49}a^{2}+\frac{1104}{7}a+\frac{978}{49}$, $a$, $\frac{333}{49}a^{11}-\frac{82}{49}a^{10}-\frac{2684}{49}a^{9}-\frac{87}{7}a^{8}+\frac{4048}{49}a^{7}-\frac{934}{7}a^{6}-161a^{5}+\frac{29531}{49}a^{4}+\frac{53066}{49}a^{3}+\frac{31869}{49}a^{2}+\frac{1255}{7}a+\frac{951}{49}$, $\frac{414}{49}a^{11}-\frac{87}{49}a^{10}-\frac{3296}{49}a^{9}-\frac{138}{7}a^{8}+\frac{4785}{49}a^{7}-\frac{1089}{7}a^{6}-\frac{1434}{7}a^{5}+\frac{35304}{49}a^{4}+\frac{68010}{49}a^{3}+\frac{45069}{49}a^{2}+\frac{2013}{7}a+\frac{1783}{49}$, $a^{11}-a^{10}-7a^{9}+3a^{8}+8a^{7}-24a^{6}-4a^{5}+84a^{4}+98a^{3}+46a^{2}+12a$, $\frac{15}{7}a^{11}-\frac{9}{7}a^{10}-\frac{118}{7}a^{9}+2a^{8}+\frac{181}{7}a^{7}-\frac{358}{7}a^{6}-\frac{235}{7}a^{5}+\frac{1422}{7}a^{4}+271a^{3}+\frac{727}{7}a^{2}+\frac{82}{7}a-\frac{4}{7}$, $\frac{101}{7}a^{11}-\frac{34}{7}a^{10}-\frac{795}{7}a^{9}-20a^{8}+\frac{1152}{7}a^{7}-\frac{1991}{7}a^{6}-\frac{2161}{7}a^{5}+\frac{8774}{7}a^{4}+2209a^{3}+\frac{9426}{7}a^{2}+\frac{2717}{7}a+\frac{296}{7}$ | 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: | \( 931.906866489 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
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Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{6}\cdot(2\pi)^{3}\cdot 931.906866489 \cdot 1}{2\cdot\sqrt{677021181018112}}\cr\approx \mathstrut & 0.284289596493 \end{aligned}\]
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
\(\Q(\sqrt{2}) \), \(\Q(\zeta_{7})^+\), 4.2.1792.1, 6.6.1229312.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Galois closure: | deg 24 |
Degree 12 sibling: | 12.0.9256148959232.2 |
Minimal sibling: | 12.0.9256148959232.2 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | ${\href{/padicField/3.12.0.1}{12} }$ | ${\href{/padicField/5.12.0.1}{12} }$ | R | ${\href{/padicField/11.6.0.1}{6} }^{2}$ | ${\href{/padicField/13.4.0.1}{4} }^{3}$ | ${\href{/padicField/17.6.0.1}{6} }{,}\,{\href{/padicField/17.3.0.1}{3} }^{2}$ | ${\href{/padicField/19.12.0.1}{12} }$ | ${\href{/padicField/23.6.0.1}{6} }^{2}$ | ${\href{/padicField/29.2.0.1}{2} }^{6}$ | ${\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.3.0.1}{3} }^{2}$ | ${\href{/padicField/37.6.0.1}{6} }^{2}$ | ${\href{/padicField/41.2.0.1}{2} }^{3}{,}\,{\href{/padicField/41.1.0.1}{1} }^{6}$ | ${\href{/padicField/43.2.0.1}{2} }^{6}$ | ${\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.3.0.1}{3} }^{2}$ | ${\href{/padicField/53.6.0.1}{6} }^{2}$ | ${\href{/padicField/59.12.0.1}{12} }$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.12.24.318 | $x^{12} + 10 x^{10} + 12 x^{9} + 110 x^{8} + 80 x^{7} + 752 x^{6} + 512 x^{5} + 1636 x^{4} + 1504 x^{3} + 1224 x^{2} + 1008 x - 648$ | $4$ | $3$ | $24$ | $C_6\times C_2$ | $[2, 3]^{3}$ |
\(7\) | 7.3.2.2 | $x^{3} + 7$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ |
7.3.2.2 | $x^{3} + 7$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ | |
7.6.5.5 | $x^{6} + 7$ | $6$ | $1$ | $5$ | $C_6$ | $[\ ]_{6}$ |