Normalized defining polynomial
\( x^{12} - 14x^{10} + 98x^{8} - 364x^{6} + 784x^{4} - 784x^{2} + 392 \)
Invariants
Degree: | $12$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[0, 6]$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2}a^{4}$, $\frac{1}{2}a^{5}$, $\frac{1}{14}a^{6}$, $\frac{1}{14}a^{7}$, $\frac{1}{28}a^{8}$, $\frac{1}{28}a^{9}$, $\frac{1}{1148}a^{10}+\frac{5}{1148}a^{8}-\frac{3}{287}a^{6}-\frac{13}{82}a^{4}+\frac{7}{41}a^{2}-\frac{18}{41}$, $\frac{1}{1148}a^{11}+\frac{5}{1148}a^{9}-\frac{3}{287}a^{7}-\frac{13}{82}a^{5}+\frac{7}{41}a^{3}-\frac{18}{41}a$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{2}$, which has order $2$
Unit group
Rank: | $5$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
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Torsion generator: | \( -\frac{1}{164} a^{10} + \frac{22}{287} a^{8} - \frac{143}{287} a^{6} + \frac{66}{41} a^{4} - \frac{131}{41} a^{2} + \frac{85}{41} \) (order $4$) | 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: | $\frac{1}{164}a^{10}-\frac{22}{287}a^{8}+\frac{35}{82}a^{6}-\frac{91}{82}a^{4}+\frac{49}{41}a^{2}-\frac{3}{41}$, $\frac{3}{574}a^{10}-\frac{13}{287}a^{8}+\frac{64}{287}a^{6}-\frac{37}{82}a^{4}+\frac{42}{41}a^{2}-\frac{26}{41}$, $\frac{1}{82}a^{11}-\frac{4}{287}a^{10}-\frac{44}{287}a^{9}+\frac{62}{287}a^{8}+\frac{286}{287}a^{7}-\frac{121}{82}a^{6}-\frac{132}{41}a^{5}+\frac{413}{82}a^{4}+\frac{221}{41}a^{3}-\frac{358}{41}a^{2}-\frac{88}{41}a+\frac{206}{41}$, $\frac{43}{1148}a^{11}+\frac{11}{287}a^{10}-\frac{141}{287}a^{9}-\frac{559}{1148}a^{8}+\frac{1833}{574}a^{7}+\frac{261}{82}a^{6}-\frac{423}{41}a^{5}-\frac{859}{82}a^{4}+\frac{711}{41}a^{3}+\frac{800}{41}a^{2}-\frac{282}{41}a-\frac{423}{41}$, $\frac{15}{1148}a^{11}+\frac{11}{1148}a^{10}-\frac{53}{287}a^{9}-\frac{191}{1148}a^{8}+\frac{689}{574}a^{7}+\frac{48}{41}a^{6}-\frac{159}{41}a^{5}-\frac{174}{41}a^{4}+\frac{228}{41}a^{3}+\frac{241}{41}a^{2}-\frac{106}{41}a-\frac{116}{41}$ | 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: | \( 2070.29507474 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 2070.29507474 \cdot 2}{4\cdot\sqrt{37913186137014272}}\cr\approx \mathstrut & 0.327104351696 \end{aligned}\]
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
\(\Q(\sqrt{-1}) \), \(\Q(\zeta_{7})^+\), 4.0.25088.2, 6.0.153664.1 |
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: | 12.6.303305489096114176.2 |
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 | ${\href{/padicField/3.12.0.1}{12} }$ | ${\href{/padicField/5.6.0.1}{6} }{,}\,{\href{/padicField/5.3.0.1}{3} }^{2}$ | R | ${\href{/padicField/11.12.0.1}{12} }$ | ${\href{/padicField/13.2.0.1}{2} }^{3}{,}\,{\href{/padicField/13.1.0.1}{1} }^{6}$ | ${\href{/padicField/17.3.0.1}{3} }^{4}$ | ${\href{/padicField/19.12.0.1}{12} }$ | ${\href{/padicField/23.6.0.1}{6} }^{2}$ | ${\href{/padicField/29.2.0.1}{2} }^{3}{,}\,{\href{/padicField/29.1.0.1}{1} }^{6}$ | ${\href{/padicField/31.6.0.1}{6} }^{2}$ | ${\href{/padicField/37.6.0.1}{6} }{,}\,{\href{/padicField/37.3.0.1}{3} }^{2}$ | ${\href{/padicField/41.2.0.1}{2} }^{6}$ | ${\href{/padicField/43.4.0.1}{4} }^{3}$ | ${\href{/padicField/47.6.0.1}{6} }^{2}$ | ${\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.3.0.1}{3} }^{2}$ | ${\href{/padicField/59.12.0.1}{12} }$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.12.27.303 | $x^{12} + 6 x^{10} + 8 x^{9} + 570 x^{8} + 256 x^{7} + 4192 x^{6} + 8832 x^{5} + 9540 x^{4} + 3072 x^{3} + 600 x^{2} + 800 x + 1000$ | $4$ | $3$ | $27$ | $D_4 \times C_3$ | $[2, 3, 7/2]^{3}$ |
\(7\) | 7.12.10.1 | $x^{12} + 36 x^{11} + 558 x^{10} + 4860 x^{9} + 26055 x^{8} + 88776 x^{7} + 193010 x^{6} + 266580 x^{5} + 237645 x^{4} + 153900 x^{3} + 137808 x^{2} + 210600 x + 184108$ | $6$ | $2$ | $10$ | $C_6\times C_2$ | $[\ ]_{6}^{2}$ |