Normalized defining polynomial
\( x^{12} - 3 x^{11} + 5 x^{10} - 13 x^{9} + 36 x^{8} - 72 x^{7} + 115 x^{6} - 150 x^{5} + 144 x^{4} + \cdots + 1 \)
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: | \(27825593350009\) \(\medspace = 7^{8}\cdot 13^{6}\) | 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: | \(13.19\) | 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: | $7^{2/3}13^{1/2}\approx 13.193814370085985$ | ||
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: | \(\Q\) | ||
$\card{ \Gal(K/\Q) }$: | $12$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
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This field is Galois over $\Q$. | |||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{7}-\frac{1}{2}$, $\frac{1}{2}a^{8}-\frac{1}{2}a$, $\frac{1}{4}a^{9}-\frac{1}{4}a^{8}-\frac{1}{4}a^{6}+\frac{1}{4}a^{5}+\frac{1}{4}a^{4}+\frac{1}{4}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{4}$, $\frac{1}{4}a^{10}-\frac{1}{4}a^{8}-\frac{1}{4}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{4}a^{3}+\frac{1}{4}a-\frac{1}{4}$, $\frac{1}{28}a^{11}-\frac{1}{14}a^{9}+\frac{1}{14}a^{8}+\frac{5}{28}a^{6}+\frac{11}{28}a^{5}-\frac{3}{7}a^{4}-\frac{11}{28}a^{3}+\frac{1}{28}a^{2}+\frac{13}{28}a-\frac{5}{28}$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $2$ |
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: | \( -1 \) (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: | $\frac{47}{28}a^{11}-4a^{10}+\frac{165}{28}a^{9}-\frac{501}{28}a^{8}+49a^{7}-\frac{629}{7}a^{6}+\frac{943}{7}a^{5}-\frac{4561}{28}a^{4}+\frac{1845}{14}a^{3}-\frac{1675}{28}a^{2}+\frac{401}{28}a+\frac{5}{14}$, $\frac{47}{28}a^{11}-4a^{10}+\frac{165}{28}a^{9}-\frac{501}{28}a^{8}+49a^{7}-\frac{629}{7}a^{6}+\frac{943}{7}a^{5}-\frac{4561}{28}a^{4}+\frac{1845}{14}a^{3}-\frac{1675}{28}a^{2}+\frac{401}{28}a-\frac{9}{14}$, $\frac{29}{28}a^{11}-\frac{15}{4}a^{10}+\frac{187}{28}a^{9}-\frac{108}{7}a^{8}+\frac{175}{4}a^{7}-\frac{648}{7}a^{6}+\frac{2109}{14}a^{5}-\frac{5633}{28}a^{4}+\frac{5659}{28}a^{3}-\frac{3709}{28}a^{2}+\frac{348}{7}a-\frac{229}{28}$, $\frac{11}{7}a^{11}-\frac{19}{4}a^{10}+\frac{213}{28}a^{9}-\frac{139}{7}a^{8}+\frac{223}{4}a^{7}-\frac{3105}{28}a^{6}+\frac{4859}{28}a^{5}-\frac{6205}{28}a^{4}+\frac{2873}{14}a^{3}-\frac{1651}{14}a^{2}+\frac{1041}{28}a-\frac{41}{7}$, $a$ | 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: | \( 39.6447347937 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 39.6447347937 \cdot 2}{2\cdot\sqrt{27825593350009}}\cr\approx \mathstrut & 0.462426282938 \end{aligned}\]
Galois group
A solvable group of order 12 |
The 4 conjugacy class representatives for $A_4$ |
Character table for $A_4$ |
Intermediate fields
\(\Q(\zeta_{7})^+\), 4.0.8281.1 x4, 6.2.405769.1 x3 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 4 sibling: | 4.0.8281.1 |
Degree 6 sibling: | 6.2.405769.1 |
Minimal sibling: | 4.0.8281.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | ${\href{/padicField/2.3.0.1}{3} }^{4}$ | ${\href{/padicField/3.3.0.1}{3} }^{4}$ | ${\href{/padicField/5.3.0.1}{3} }^{4}$ | R | ${\href{/padicField/11.3.0.1}{3} }^{4}$ | R | ${\href{/padicField/17.3.0.1}{3} }^{4}$ | ${\href{/padicField/19.3.0.1}{3} }^{4}$ | ${\href{/padicField/23.3.0.1}{3} }^{4}$ | ${\href{/padicField/29.2.0.1}{2} }^{6}$ | ${\href{/padicField/31.3.0.1}{3} }^{4}$ | ${\href{/padicField/37.3.0.1}{3} }^{4}$ | ${\href{/padicField/41.2.0.1}{2} }^{6}$ | ${\href{/padicField/43.2.0.1}{2} }^{6}$ | ${\href{/padicField/47.3.0.1}{3} }^{4}$ | ${\href{/padicField/53.3.0.1}{3} }^{4}$ | ${\href{/padicField/59.3.0.1}{3} }^{4}$ |
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 |
---|---|---|---|---|---|---|---|
\(7\) | 7.3.2.2 | $x^{3} + 7$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ |
7.3.2.2 | $x^{3} + 7$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ | |
7.3.2.2 | $x^{3} + 7$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ | |
7.3.2.2 | $x^{3} + 7$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ | |
\(13\) | 13.4.2.1 | $x^{4} + 284 x^{3} + 21754 x^{2} + 225780 x + 59193$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
13.4.2.1 | $x^{4} + 284 x^{3} + 21754 x^{2} + 225780 x + 59193$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
13.4.2.1 | $x^{4} + 284 x^{3} + 21754 x^{2} + 225780 x + 59193$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |