Normalized defining polynomial
\( x^{12} + 18 x^{10} - 12 x^{9} + 189 x^{8} - 288 x^{7} + 1458 x^{6} - 2796 x^{5} + 7092 x^{4} + \cdots + 10185 \)
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: | \(84966594432586481664\) \(\medspace = 2^{24}\cdot 3^{16}\cdot 7^{6}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(45.79\) | 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: | $2^{21/8}3^{37/24}7^{2/3}\approx 122.79038648593725$ | ||
Ramified primes: | \(2\), \(3\), \(7\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q\) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{35\!\cdots\!57}a^{11}-\frac{16\!\cdots\!52}{35\!\cdots\!57}a^{10}-\frac{12\!\cdots\!33}{35\!\cdots\!57}a^{9}-\frac{93\!\cdots\!80}{35\!\cdots\!57}a^{8}-\frac{134377992424490}{834330308291599}a^{7}-\frac{16\!\cdots\!04}{35\!\cdots\!57}a^{6}-\frac{648567382710480}{51\!\cdots\!51}a^{5}+\frac{17\!\cdots\!98}{35\!\cdots\!57}a^{4}-\frac{17\!\cdots\!65}{35\!\cdots\!57}a^{3}-\frac{20\!\cdots\!21}{51\!\cdots\!51}a^{2}+\frac{17\!\cdots\!80}{51\!\cdots\!51}a+\frac{10\!\cdots\!38}{51\!\cdots\!51}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{2}$, which has order $2$ (assuming GRH)
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: | $\frac{137365392995000}{35\!\cdots\!57}a^{11}+\frac{57202180009932}{35\!\cdots\!57}a^{10}+\frac{15\!\cdots\!90}{35\!\cdots\!57}a^{9}-\frac{441930182552623}{35\!\cdots\!57}a^{8}+\frac{331137736505143}{834330308291599}a^{7}-\frac{17\!\cdots\!03}{35\!\cdots\!57}a^{6}+\frac{10\!\cdots\!83}{51\!\cdots\!51}a^{5}-\frac{97\!\cdots\!80}{35\!\cdots\!57}a^{4}+\frac{11\!\cdots\!38}{35\!\cdots\!57}a^{3}+\frac{33\!\cdots\!70}{51\!\cdots\!51}a^{2}-\frac{58\!\cdots\!77}{51\!\cdots\!51}a+\frac{50\!\cdots\!94}{51\!\cdots\!51}$, $\frac{244996352335894}{35\!\cdots\!57}a^{11}+\frac{132790227207898}{35\!\cdots\!57}a^{10}+\frac{29\!\cdots\!02}{35\!\cdots\!57}a^{9}-\frac{678320973391759}{35\!\cdots\!57}a^{8}+\frac{618853468592253}{834330308291599}a^{7}-\frac{33\!\cdots\!70}{35\!\cdots\!57}a^{6}+\frac{20\!\cdots\!31}{51\!\cdots\!51}a^{5}-\frac{20\!\cdots\!00}{35\!\cdots\!57}a^{4}+\frac{30\!\cdots\!27}{35\!\cdots\!57}a^{3}-\frac{25\!\cdots\!47}{51\!\cdots\!51}a^{2}-\frac{51\!\cdots\!35}{51\!\cdots\!51}a+\frac{56\!\cdots\!51}{51\!\cdots\!51}$, $\frac{61842024969527}{35\!\cdots\!57}a^{11}-\frac{68332426662849}{35\!\cdots\!57}a^{10}+\frac{780560056989309}{35\!\cdots\!57}a^{9}-\frac{13\!\cdots\!43}{35\!\cdots\!57}a^{8}+\frac{182667548672244}{834330308291599}a^{7}-\frac{19\!\cdots\!68}{35\!\cdots\!57}a^{6}+\frac{83\!\cdots\!77}{51\!\cdots\!51}a^{5}-\frac{13\!\cdots\!29}{35\!\cdots\!57}a^{4}+\frac{22\!\cdots\!74}{35\!\cdots\!57}a^{3}-\frac{46\!\cdots\!38}{51\!\cdots\!51}a^{2}+\frac{35\!\cdots\!49}{51\!\cdots\!51}a-\frac{16\!\cdots\!19}{51\!\cdots\!51}$, $\frac{18712358906398}{35\!\cdots\!57}a^{11}+\frac{1298278690060}{35\!\cdots\!57}a^{10}+\frac{314397901023770}{35\!\cdots\!57}a^{9}+\frac{161304746171832}{35\!\cdots\!57}a^{8}+\frac{54792221076682}{834330308291599}a^{7}+\frac{11\!\cdots\!19}{35\!\cdots\!57}a^{6}+\frac{17\!\cdots\!49}{51\!\cdots\!51}a^{5}-\frac{49\!\cdots\!21}{35\!\cdots\!57}a^{4}+\frac{46\!\cdots\!60}{35\!\cdots\!57}a^{3}-\frac{81\!\cdots\!20}{51\!\cdots\!51}a^{2}+\frac{12\!\cdots\!94}{51\!\cdots\!51}a-\frac{16\!\cdots\!29}{51\!\cdots\!51}$, $\frac{3832778393550}{51\!\cdots\!51}a^{11}+\frac{5280664539066}{51\!\cdots\!51}a^{10}+\frac{58587539318928}{51\!\cdots\!51}a^{9}+\frac{59708305665525}{51\!\cdots\!51}a^{8}+\frac{13189087323922}{119190044041657}a^{7}+\frac{7161541192178}{51\!\cdots\!51}a^{6}+\frac{26\!\cdots\!04}{51\!\cdots\!51}a^{5}-\frac{13\!\cdots\!79}{51\!\cdots\!51}a^{4}+\frac{30\!\cdots\!76}{51\!\cdots\!51}a^{3}-\frac{10\!\cdots\!45}{51\!\cdots\!51}a^{2}-\frac{14\!\cdots\!89}{51\!\cdots\!51}a+\frac{18\!\cdots\!53}{51\!\cdots\!51}$ (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)]
| |
Regulator: | \( 130869.789094 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 130869.789094 \cdot 2}{2\cdot\sqrt{84966594432586481664}}\cr\approx \mathstrut & 0.873563554777 \end{aligned}\] (assuming GRH)
Galois group
A non-solvable group of order 95040 |
The 15 conjugacy class representatives for $M_{12}$ |
Character table for $M_{12}$ |
Intermediate fields
The extension is primitive: there are no intermediate fields between this field and $\Q$. |
Sibling fields
Degree 12 sibling: | data not computed |
Minimal sibling: | 12.0.9440732714731831296.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | R | ${\href{/padicField/5.8.0.1}{8} }{,}\,{\href{/padicField/5.2.0.1}{2} }{,}\,{\href{/padicField/5.1.0.1}{1} }^{2}$ | R | ${\href{/padicField/11.3.0.1}{3} }^{4}$ | ${\href{/padicField/13.6.0.1}{6} }^{2}$ | ${\href{/padicField/17.8.0.1}{8} }{,}\,{\href{/padicField/17.4.0.1}{4} }$ | ${\href{/padicField/19.8.0.1}{8} }{,}\,{\href{/padicField/19.4.0.1}{4} }$ | ${\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.3.0.1}{3} }{,}\,{\href{/padicField/23.2.0.1}{2} }{,}\,{\href{/padicField/23.1.0.1}{1} }$ | ${\href{/padicField/29.8.0.1}{8} }{,}\,{\href{/padicField/29.2.0.1}{2} }{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.6.0.1}{6} }^{2}$ | ${\href{/padicField/37.6.0.1}{6} }{,}\,{\href{/padicField/37.3.0.1}{3} }{,}\,{\href{/padicField/37.2.0.1}{2} }{,}\,{\href{/padicField/37.1.0.1}{1} }$ | ${\href{/padicField/41.5.0.1}{5} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.4.0.1}{4} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }^{4}$ | ${\href{/padicField/47.11.0.1}{11} }{,}\,{\href{/padicField/47.1.0.1}{1} }$ | ${\href{/padicField/53.11.0.1}{11} }{,}\,{\href{/padicField/53.1.0.1}{1} }$ | ${\href{/padicField/59.11.0.1}{11} }{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.12.24.239 | $x^{12} + 22 x^{10} - 4 x^{9} - 158 x^{8} - 160 x^{7} - 168 x^{6} + 80 x^{5} + 1292 x^{4} + 3104 x^{3} + 4136 x^{2} + 3216 x + 5048$ | $4$ | $3$ | $24$ | 12T60 | $[2, 2, 3, 3]^{6}$ |
\(3\) | 3.12.16.48 | $x^{12} + 6 x^{7} + 3 x^{6} + 3 x^{5} + 6$ | $12$ | $1$ | $16$ | 12T84 | $[13/8, 13/8]_{8}^{2}$ |
\(7\) | 7.3.2.1 | $x^{3} + 14$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ |
7.3.2.3 | $x^{3} + 21$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ | |
7.3.0.1 | $x^{3} + 6 x^{2} + 4$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
7.3.2.2 | $x^{3} + 7$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ |