Normalized defining polynomial
\( x^{12} - 2x^{11} + x^{10} - 3x^{9} + 3x^{8} + 3x^{7} - x^{6} - 5x^{5} - 15x^{4} + 10x^{3} - 3x^{2} - 3x + 1 \)
Invariants
Degree: | $12$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[4, 4]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(8067775586689\) \(\medspace = 7^{10}\cdot 13^{4}\) | 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: | \(11.90\) | 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: | $7^{5/6}13^{1/2}\approx 18.248200448594663$ | ||
Ramified primes: | \(7\), \(13\) | 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: | \(\Q\) | ||
$\card{ \Aut(K/\Q) }$: | $4$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{674}a^{10}-\frac{32}{337}a^{9}-\frac{60}{337}a^{8}-\frac{65}{337}a^{7}-\frac{17}{674}a^{6}+\frac{167}{674}a^{5}-\frac{77}{337}a^{4}-\frac{331}{674}a^{3}+\frac{70}{337}a^{2}+\frac{159}{674}a-\frac{161}{337}$, $\frac{1}{674}a^{11}+\frac{165}{674}a^{9}+\frac{139}{337}a^{8}-\frac{249}{674}a^{7}-\frac{247}{674}a^{6}+\frac{87}{674}a^{5}+\frac{130}{337}a^{4}-\frac{75}{337}a^{3}+\frac{10}{337}a^{2}-\frac{128}{337}a-\frac{51}{674}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$
Unit group
Rank: | $7$ | 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: | $\frac{159}{337}a^{11}-\frac{279}{337}a^{10}+\frac{225}{674}a^{9}-\frac{502}{337}a^{8}+\frac{386}{337}a^{7}+\frac{518}{337}a^{6}+\frac{195}{674}a^{5}-\frac{1573}{674}a^{4}-\frac{2608}{337}a^{3}+\frac{1369}{674}a^{2}-\frac{478}{337}a-\frac{661}{674}$, $\frac{82}{337}a^{11}-\frac{167}{337}a^{10}+\frac{245}{674}a^{9}-\frac{300}{337}a^{8}+\frac{281}{337}a^{7}+\frac{109}{337}a^{6}-\frac{59}{674}a^{5}-\frac{621}{674}a^{4}-\frac{1170}{337}a^{3}+\frac{1341}{674}a^{2}-\frac{702}{337}a-\frac{231}{674}$, $\frac{29}{674}a^{11}-\frac{20}{337}a^{10}-\frac{69}{674}a^{9}+\frac{28}{337}a^{8}+\frac{1}{674}a^{7}+\frac{257}{674}a^{6}-\frac{113}{674}a^{5}-\frac{227}{337}a^{4}-\frac{273}{337}a^{3}+\frac{186}{337}a^{2}+\frac{522}{337}a-\frac{57}{674}$, $\frac{57}{674}a^{11}-\frac{85}{674}a^{10}+\frac{17}{674}a^{9}-\frac{120}{337}a^{8}+\frac{227}{674}a^{7}+\frac{86}{337}a^{6}+\frac{100}{337}a^{5}-\frac{199}{337}a^{4}-\frac{1309}{674}a^{3}+\frac{12}{337}a^{2}+\frac{201}{674}a+\frac{873}{674}$, $\frac{36}{337}a^{11}-\frac{6}{337}a^{10}-\frac{79}{337}a^{9}-\frac{56}{337}a^{8}-\frac{96}{337}a^{7}+\frac{309}{337}a^{6}+\frac{108}{337}a^{5}-\frac{163}{337}a^{4}-\frac{1055}{337}a^{3}-\frac{794}{337}a^{2}+\frac{277}{337}a-\frac{241}{337}$, $\frac{95}{337}a^{11}-\frac{241}{674}a^{10}-\frac{69}{674}a^{9}-\frac{244}{337}a^{8}+\frac{98}{337}a^{7}+\frac{977}{674}a^{6}+\frac{105}{337}a^{5}-\frac{769}{674}a^{4}-\frac{3997}{674}a^{3}+\frac{53}{674}a^{2}+\frac{661}{674}a-\frac{1173}{674}$, $\frac{77}{674}a^{11}-\frac{27}{674}a^{10}-\frac{29}{337}a^{9}-\frac{146}{337}a^{8}-\frac{161}{674}a^{7}+\frac{156}{337}a^{6}+\frac{505}{674}a^{5}+\frac{251}{674}a^{4}-\frac{1939}{674}a^{3}-\frac{1903}{674}a^{2}-\frac{1089}{674}a+\frac{193}{337}$ | 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: | \( 47.1747110525 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{4}\cdot 47.1747110525 \cdot 1}{2\cdot\sqrt{8067775586689}}\cr\approx \mathstrut & 0.207081739539 \end{aligned}\]
Galois group
$C_2^2\times A_4$ (as 12T26):
A solvable group of order 48 |
The 16 conjugacy class representatives for $C_2^2 \times A_4$ |
Character table for $C_2^2 \times A_4$ |
Intermediate fields
\(\Q(\zeta_{7})^+\), 6.2.31213.1, 6.4.218491.1, 6.4.2840383.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 12 siblings: | 12.0.47738317081.1, 12.8.1363454074150441.1, 12.0.1363454074150441.2 |
Degree 16 sibling: | data not computed |
Degree 24 siblings: | data not computed |
Arithmetically equvalently sibling: | 12.4.8067775586689.2 |
Minimal sibling: | 12.0.47738317081.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | ${\href{/padicField/2.6.0.1}{6} }^{2}$ | ${\href{/padicField/3.6.0.1}{6} }^{2}$ | ${\href{/padicField/5.6.0.1}{6} }^{2}$ | R | ${\href{/padicField/11.6.0.1}{6} }^{2}$ | R | ${\href{/padicField/17.6.0.1}{6} }^{2}$ | ${\href{/padicField/19.6.0.1}{6} }^{2}$ | ${\href{/padicField/23.3.0.1}{3} }^{4}$ | ${\href{/padicField/29.2.0.1}{2} }^{6}$ | ${\href{/padicField/31.6.0.1}{6} }^{2}$ | ${\href{/padicField/37.6.0.1}{6} }^{2}$ | ${\href{/padicField/41.2.0.1}{2} }^{4}{,}\,{\href{/padicField/41.1.0.1}{1} }^{4}$ | ${\href{/padicField/43.2.0.1}{2} }^{6}$ | ${\href{/padicField/47.6.0.1}{6} }^{2}$ | ${\href{/padicField/53.3.0.1}{3} }^{4}$ | ${\href{/padicField/59.6.0.1}{6} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(7\) | 7.12.10.1 | $x^{12} + 36 x^{11} + 558 x^{10} + 4860 x^{9} + 26055 x^{8} + 88776 x^{7} + 193010 x^{6} + 266580 x^{5} + 237645 x^{4} + 153900 x^{3} + 137808 x^{2} + 210600 x + 184108$ | $6$ | $2$ | $10$ | $C_6\times C_2$ | $[\ ]_{6}^{2}$ |
\(13\) | 13.2.1.2 | $x^{2} + 26$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
13.2.1.1 | $x^{2} + 13$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
13.2.1.2 | $x^{2} + 26$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
13.2.0.1 | $x^{2} + 12 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
13.2.0.1 | $x^{2} + 12 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
13.2.1.1 | $x^{2} + 13$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |