Normalized defining polynomial
\( x^{12} - 2x^{10} + 6x^{8} + 4x^{6} + 104 \)
Invariants
Degree: | $12$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[0, 6]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(49834102882304\) \(\medspace = 2^{27}\cdot 13^{5}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(13.85\) | 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))
| |
Galois root discriminant: | $2^{203/64}13^{5/6}\approx 76.40392057046596$ | ||
Ramified primes: | \(2\), \(13\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{26}) \) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
This field is not Galois over $\Q$. | |||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2}a^{4}$, $\frac{1}{2}a^{5}$, $\frac{1}{2}a^{6}$, $\frac{1}{2}a^{7}$, $\frac{1}{4}a^{8}$, $\frac{1}{4}a^{9}$, $\frac{1}{8612}a^{10}+\frac{19}{8612}a^{8}-\frac{437}{2153}a^{6}+\frac{1025}{4306}a^{4}+\frac{1074}{2153}a^{2}+\frac{1024}{2153}$, $\frac{1}{8612}a^{11}+\frac{19}{8612}a^{9}-\frac{437}{2153}a^{7}+\frac{1025}{4306}a^{5}+\frac{1074}{2153}a^{3}+\frac{1024}{2153}a$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$
Unit group
Rank: | $5$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
Torsion generator: | \( -\frac{103}{8612} a^{10} + \frac{49}{2153} a^{8} - \frac{202}{2153} a^{6} - \frac{39}{2153} a^{4} - \frac{819}{2153} a^{2} + \frac{25}{2153} \) (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)
| |
Fundamental units: | $\frac{29}{2153}a^{10}+\frac{51}{8612}a^{8}-\frac{193}{4306}a^{6}+\frac{485}{4306}a^{4}-\frac{290}{2153}a^{2}-\frac{1784}{2153}$, $\frac{53}{4306}a^{10}-\frac{139}{8612}a^{8}-\frac{65}{4306}a^{6}+\frac{500}{2153}a^{4}-\frac{265}{2153}a^{2}-\frac{1259}{2153}$, $\frac{27}{2153}a^{11}+\frac{5}{2153}a^{10}-\frac{101}{8612}a^{9}+\frac{95}{2153}a^{8}+\frac{170}{2153}a^{7}-\frac{128}{2153}a^{6}+\frac{897}{4306}a^{5}+\frac{1123}{4306}a^{4}-\frac{270}{2153}a^{3}-\frac{50}{2153}a^{2}+\frac{789}{2153}a-\frac{1050}{2153}$, $\frac{29}{2153}a^{11}+\frac{101}{8612}a^{10}+\frac{51}{8612}a^{9}-\frac{117}{4306}a^{8}-\frac{193}{4306}a^{7}-\frac{1}{4306}a^{6}+\frac{485}{4306}a^{5}+\frac{181}{4306}a^{4}-\frac{290}{2153}a^{3}-\frac{1329}{2153}a^{2}-\frac{1784}{2153}a-\frac{2073}{2153}$, $\frac{217}{8612}a^{11}+\frac{389}{8612}a^{10}-\frac{183}{8612}a^{9}-\frac{1221}{8612}a^{8}-\frac{97}{2153}a^{7}+\frac{94}{2153}a^{6}+\frac{333}{2153}a^{5}+\frac{210}{2153}a^{4}+\frac{534}{2153}a^{3}+\frac{104}{2153}a^{2}+\frac{449}{2153}a-\frac{2122}{2153}$ | sage: UK.fundamental_units()
gp: K.fu
magma: [K|fUK(g): g in Generators(UK)];
oscar: [K(fUK(a)) for a in gens(UK)]
| |
Regulator: | \( 115.500745079 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 115.500745079 \cdot 1}{4\cdot\sqrt{49834102882304}}\cr\approx \mathstrut & 0.251675350667 \end{aligned}\]
Galois group
$A_4^2:D_4$ (as 12T208):
A solvable group of order 1152 |
The 44 conjugacy class representatives for $A_4^2:D_4$ |
Character table for $A_4^2:D_4$ is not computed |
Intermediate fields
\(\Q(\sqrt{-1}) \), 6.0.10816.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 12 sibling: | data not computed |
Degree 16 siblings: | data not computed |
Degree 24 siblings: | data not computed |
Degree 32 sibling: | data not computed |
Degree 36 siblings: | data not computed |
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 | ${\href{/padicField/3.12.0.1}{12} }$ | ${\href{/padicField/5.6.0.1}{6} }^{2}$ | ${\href{/padicField/7.12.0.1}{12} }$ | ${\href{/padicField/11.6.0.1}{6} }^{2}$ | R | ${\href{/padicField/17.3.0.1}{3} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }^{2}{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ | ${\href{/padicField/19.6.0.1}{6} }^{2}$ | ${\href{/padicField/23.6.0.1}{6} }^{2}$ | ${\href{/padicField/29.3.0.1}{3} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }{,}\,{\href{/padicField/29.1.0.1}{1} }^{4}$ | ${\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.2.0.1}{2} }^{4}$ | ${\href{/padicField/37.3.0.1}{3} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.2.0.1}{2} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.12.0.1}{12} }$ | ${\href{/padicField/47.4.0.1}{4} }{,}\,{\href{/padicField/47.2.0.1}{2} }^{4}$ | ${\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.3.0.1}{3} }^{2}$ | ${\href{/padicField/59.6.0.1}{6} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.12.27.38 | $x^{12} + 30 x^{10} + 128 x^{9} + 74 x^{8} + 32 x^{7} + 2912 x^{6} + 2368 x^{5} + 7364 x^{4} + 704 x^{3} + 3768 x^{2} + 552$ | $4$ | $3$ | $27$ | 12T134 | $[2, 2, 2, 3, 3, 3, 7/2]^{3}$ |
\(13\) | 13.3.0.1 | $x^{3} + 2 x + 11$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ |
13.3.0.1 | $x^{3} + 2 x + 11$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
13.6.5.4 | $x^{6} + 26$ | $6$ | $1$ | $5$ | $C_6$ | $[\ ]_{6}$ |