Normalized defining polynomial
\( x^{12} - 16x^{10} + 88x^{8} - 204x^{6} + 212x^{4} - 88x^{2} + 8 \)
Invariants
Degree: | $12$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[12, 0]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(37913186137014272\) \(\medspace = 2^{27}\cdot 7^{10}\) | 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: | \(24.07\) | 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: | $2^{11/4}7^{5/6}\approx 34.04715710793443$ | ||
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: | \(\Q(\sqrt{2}) \) | ||
$\card{ \Aut(K/\Q) }$: | $6$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2}a^{4}$, $\frac{1}{2}a^{5}$, $\frac{1}{2}a^{6}$, $\frac{1}{2}a^{7}$, $\frac{1}{4}a^{8}$, $\frac{1}{4}a^{9}$, $\frac{1}{4}a^{10}$, $\frac{1}{4}a^{11}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
Rank: | $11$ | 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: | $\frac{1}{2}a^{10}-\frac{29}{4}a^{8}+33a^{6}-51a^{4}+24a^{2}$, $\frac{3}{4}a^{10}-11a^{8}+\frac{103}{2}a^{6}-\frac{173}{2}a^{4}+52a^{2}-7$, $\frac{1}{4}a^{8}-\frac{7}{2}a^{6}+\frac{29}{2}a^{4}-16a^{2}+2$, $\frac{3}{2}a^{10}-\frac{87}{4}a^{8}+\frac{199}{2}a^{6}-\frac{317}{2}a^{4}+88a^{2}-11$, $2a^{10}-29a^{8}+\frac{265}{2}a^{6}-\frac{419}{2}a^{4}+112a^{2}-12$, $\frac{5}{4}a^{11}-\frac{1}{2}a^{10}-18a^{9}+\frac{29}{4}a^{8}+81a^{7}-33a^{6}-\frac{245}{2}a^{5}+51a^{4}+56a^{3}-24a^{2}-a+1$, $\frac{3}{4}a^{11}-\frac{3}{4}a^{10}-11a^{9}+11a^{8}+\frac{103}{2}a^{7}-\frac{103}{2}a^{6}-\frac{173}{2}a^{5}+\frac{173}{2}a^{4}+52a^{3}-52a^{2}-6a+6$, $\frac{1}{2}a^{11}+\frac{3}{4}a^{10}-\frac{29}{4}a^{9}-11a^{8}+33a^{7}+\frac{103}{2}a^{6}-\frac{101}{2}a^{5}-\frac{173}{2}a^{4}+20a^{3}+52a^{2}+4a-7$, $\frac{5}{4}a^{11}+\frac{5}{4}a^{10}-18a^{9}-\frac{73}{4}a^{8}+81a^{7}+\frac{169}{2}a^{6}-\frac{245}{2}a^{5}-\frac{275}{2}a^{4}+56a^{3}+76a^{2}-a-8$, $\frac{1}{2}a^{11}-\frac{29}{4}a^{9}+\frac{1}{4}a^{8}+33a^{7}-\frac{7}{2}a^{6}-\frac{101}{2}a^{5}+15a^{4}+20a^{3}-20a^{2}+5a+6$, $\frac{1}{2}a^{11}+\frac{1}{2}a^{10}-\frac{29}{4}a^{9}-\frac{29}{4}a^{8}+33a^{7}+33a^{6}-\frac{101}{2}a^{5}-51a^{4}+20a^{3}+24a^{2}+4a$ (assuming GRH) | 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: | \( 18736.3891264 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
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Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{12}\cdot(2\pi)^{0}\cdot 18736.3891264 \cdot 1}{2\cdot\sqrt{37913186137014272}}\cr\approx \mathstrut & 0.197070086961 \end{aligned}\] (assuming GRH)
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
\(\Q(\sqrt{7}) \), \(\Q(\zeta_{7})^+\), 4.4.25088.1, \(\Q(\zeta_{28})^+\) |
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: | 12.12.43329355585159168.1 |
Minimal sibling: | This field is its own minimal sibling |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | ${\href{/padicField/3.6.0.1}{6} }{,}\,{\href{/padicField/3.3.0.1}{3} }^{2}$ | ${\href{/padicField/5.12.0.1}{12} }$ | R | ${\href{/padicField/11.12.0.1}{12} }$ | ${\href{/padicField/13.4.0.1}{4} }^{3}$ | ${\href{/padicField/17.6.0.1}{6} }^{2}$ | ${\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.3.0.1}{3} }^{2}$ | ${\href{/padicField/23.6.0.1}{6} }^{2}$ | ${\href{/padicField/29.2.0.1}{2} }^{3}{,}\,{\href{/padicField/29.1.0.1}{1} }^{6}$ | ${\href{/padicField/31.6.0.1}{6} }^{2}$ | ${\href{/padicField/37.6.0.1}{6} }{,}\,{\href{/padicField/37.3.0.1}{3} }^{2}$ | ${\href{/padicField/41.2.0.1}{2} }^{6}$ | ${\href{/padicField/43.4.0.1}{4} }^{3}$ | ${\href{/padicField/47.6.0.1}{6} }^{2}$ | ${\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.3.0.1}{3} }^{2}$ | ${\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.3.0.1}{3} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.12.27.11 | $x^{12} - 2 x^{10} + 8 x^{9} + 314 x^{8} + 256 x^{7} - 1312 x^{6} + 2688 x^{5} + 5444 x^{4} + 3072 x^{3} - 200 x^{2} + 800 x + 1000$ | $4$ | $3$ | $27$ | $D_4 \times C_3$ | $[2, 3, 7/2]^{3}$ |
\(7\) | 7.12.10.1 | $x^{12} + 36 x^{11} + 558 x^{10} + 4860 x^{9} + 26055 x^{8} + 88776 x^{7} + 193010 x^{6} + 266580 x^{5} + 237645 x^{4} + 153900 x^{3} + 137808 x^{2} + 210600 x + 184108$ | $6$ | $2$ | $10$ | $C_6\times C_2$ | $[\ ]_{6}^{2}$ |