Normalized defining polynomial
\( x^{12} - 2 x^{11} - x^{10} + 6 x^{9} - 3 x^{8} - 4 x^{7} - 10 x^{6} + 14 x^{5} + 23 x^{4} - 64 x^{3} + \cdots + 25 \)
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: | \(41343459692544\) \(\medspace = 2^{12}\cdot 3^{6}\cdot 61^{4}\) | 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: | \(13.64\) | 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: | $2\cdot 3^{1/2}61^{2/3}\approx 53.67975352058531$ | ||
Ramified primes: | \(2\), \(3\), \(61\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q\) | ||
$\card{ \Aut(K/\Q) }$: | $6$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{207}a^{10}+\frac{1}{69}a^{9}+\frac{34}{207}a^{8}+\frac{29}{207}a^{7}-\frac{2}{69}a^{6}-\frac{25}{69}a^{5}-\frac{91}{207}a^{4}-\frac{26}{69}a^{3}+\frac{10}{23}a^{2}+\frac{68}{207}a-\frac{53}{207}$, $\frac{1}{860085}a^{11}-\frac{397}{860085}a^{10}+\frac{260689}{860085}a^{9}-\frac{310409}{860085}a^{8}-\frac{289193}{860085}a^{7}+\frac{90682}{286695}a^{6}+\frac{57856}{172017}a^{5}+\frac{75859}{860085}a^{4}+\frac{49001}{286695}a^{3}+\frac{58046}{860085}a^{2}+\frac{182024}{860085}a-\frac{65933}{172017}$
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)
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Torsion generator: | \( -\frac{34663}{860085} a^{11} + \frac{91261}{860085} a^{10} + \frac{64433}{860085} a^{9} - \frac{288583}{860085} a^{8} + \frac{76919}{860085} a^{7} + \frac{117689}{286695} a^{6} + \frac{88661}{172017} a^{5} - \frac{798217}{860085} a^{4} - \frac{22186}{12465} a^{3} + \frac{2756707}{860085} a^{2} - \frac{2299322}{860085} a + \frac{252890}{172017} \) (order $12$) | 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: | $\frac{34663}{860085}a^{11}-\frac{91261}{860085}a^{10}-\frac{64433}{860085}a^{9}+\frac{288583}{860085}a^{8}-\frac{76919}{860085}a^{7}-\frac{117689}{286695}a^{6}-\frac{88661}{172017}a^{5}+\frac{798217}{860085}a^{4}+\frac{22186}{12465}a^{3}-\frac{2756707}{860085}a^{2}+\frac{2299322}{860085}a-\frac{80873}{172017}$, $\frac{117791}{860085}a^{11}-\frac{139772}{860085}a^{10}-\frac{260761}{860085}a^{9}+\frac{601016}{860085}a^{8}+\frac{214622}{860085}a^{7}-\frac{229483}{286695}a^{6}-\frac{325147}{172017}a^{5}+\frac{1027589}{860085}a^{4}+\frac{1221661}{286695}a^{3}-\frac{5801639}{860085}a^{2}+\frac{3278389}{860085}a-\frac{92149}{172017}$, $\frac{91976}{860085}a^{11}-\frac{62657}{860085}a^{10}-\frac{193516}{860085}a^{9}+\frac{322616}{860085}a^{8}+\frac{231512}{860085}a^{7}-\frac{46408}{286695}a^{6}-\frac{261655}{172017}a^{5}-\frac{429541}{860085}a^{4}+\frac{710446}{286695}a^{3}-\frac{2849879}{860085}a^{2}+\frac{2823604}{860085}a-\frac{349381}{172017}$, $\frac{62969}{860085}a^{11}-\frac{31298}{860085}a^{10}-\frac{181879}{860085}a^{9}+\frac{134924}{860085}a^{8}+\frac{248573}{860085}a^{7}+\frac{683}{286695}a^{6}-\frac{215512}{172017}a^{5}-\frac{695179}{860085}a^{4}+\frac{630979}{286695}a^{3}-\frac{599231}{860085}a^{2}+\frac{2071786}{860085}a-\frac{197023}{172017}$, $\frac{24770}{172017}a^{11}-\frac{20411}{172017}a^{10}-\frac{54703}{172017}a^{9}+\frac{95459}{172017}a^{8}+\frac{62294}{172017}a^{7}-\frac{21466}{57339}a^{6}-\frac{353768}{172017}a^{5}+\frac{17597}{172017}a^{4}+\frac{245470}{57339}a^{3}-\frac{890936}{172017}a^{2}+\frac{733990}{172017}a-\frac{251456}{172017}$ | 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: | \( 312.3168359232439 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 312.3168359232439 \cdot 1}{12\cdot\sqrt{41343459692544}}\cr\approx \mathstrut & 0.249051953235256 \end{aligned}\]
Galois group
$C_6\times S_3$ (as 12T18):
A solvable group of order 36 |
The 18 conjugacy class representatives for $C_6\times S_3$ |
Character table for $C_6\times S_3$ |
Intermediate fields
\(\Q(\sqrt{3}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-1}) \), \(\Q(\zeta_{12})\), 6.0.238144.2 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Galois closure: | deg 36 |
Degree 18 siblings: | 18.6.13695857684932038060731857108992.1, deg 18 |
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 | R | ${\href{/padicField/5.6.0.1}{6} }{,}\,{\href{/padicField/5.2.0.1}{2} }^{3}$ | ${\href{/padicField/7.6.0.1}{6} }^{2}$ | ${\href{/padicField/11.2.0.1}{2} }^{6}$ | ${\href{/padicField/13.3.0.1}{3} }^{4}$ | ${\href{/padicField/17.6.0.1}{6} }^{2}$ | ${\href{/padicField/19.6.0.1}{6} }^{2}$ | ${\href{/padicField/23.2.0.1}{2} }^{6}$ | ${\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.2.0.1}{2} }^{3}$ | ${\href{/padicField/31.6.0.1}{6} }^{2}$ | ${\href{/padicField/37.3.0.1}{3} }^{4}$ | ${\href{/padicField/41.6.0.1}{6} }^{2}$ | ${\href{/padicField/43.6.0.1}{6} }^{2}$ | ${\href{/padicField/47.6.0.1}{6} }^{2}$ | ${\href{/padicField/53.6.0.1}{6} }^{2}$ | ${\href{/padicField/59.6.0.1}{6} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.12.12.26 | $x^{12} + 12 x^{11} + 98 x^{10} + 542 x^{9} + 2359 x^{8} + 7956 x^{7} + 21831 x^{6} + 47308 x^{5} + 82476 x^{4} + 109442 x^{3} + 112071 x^{2} + 76900 x + 33205$ | $2$ | $6$ | $12$ | $C_6\times C_2$ | $[2]^{6}$ |
\(3\) | 3.4.2.1 | $x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
3.4.2.1 | $x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
3.4.2.1 | $x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
\(61\) | 61.3.2.2 | $x^{3} + 122$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ |
61.3.0.1 | $x^{3} + 7 x + 59$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
61.3.2.2 | $x^{3} + 122$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ | |
61.3.0.1 | $x^{3} + 7 x + 59$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
* | 1.12.2t1.a.a | $1$ | $ 2^{2} \cdot 3 $ | \(\Q(\sqrt{3}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
* | 1.4.2t1.a.a | $1$ | $ 2^{2}$ | \(\Q(\sqrt{-1}) \) | $C_2$ (as 2T1) | $1$ | $-1$ |
* | 1.3.2t1.a.a | $1$ | $ 3 $ | \(\Q(\sqrt{-3}) \) | $C_2$ (as 2T1) | $1$ | $-1$ |
1.244.6t1.b.a | $1$ | $ 2^{2} \cdot 61 $ | 6.0.886133824.1 | $C_6$ (as 6T1) | $0$ | $-1$ | |
1.183.6t1.a.a | $1$ | $ 3 \cdot 61 $ | 6.0.373837707.1 | $C_6$ (as 6T1) | $0$ | $-1$ | |
1.61.3t1.a.a | $1$ | $ 61 $ | 3.3.3721.1 | $C_3$ (as 3T1) | $0$ | $1$ | |
1.61.3t1.a.b | $1$ | $ 61 $ | 3.3.3721.1 | $C_3$ (as 3T1) | $0$ | $1$ | |
1.732.6t1.a.a | $1$ | $ 2^{2} \cdot 3 \cdot 61 $ | 6.6.23925613248.1 | $C_6$ (as 6T1) | $0$ | $1$ | |
1.244.6t1.b.b | $1$ | $ 2^{2} \cdot 61 $ | 6.0.886133824.1 | $C_6$ (as 6T1) | $0$ | $-1$ | |
1.732.6t1.a.b | $1$ | $ 2^{2} \cdot 3 \cdot 61 $ | 6.6.23925613248.1 | $C_6$ (as 6T1) | $0$ | $1$ | |
1.183.6t1.a.b | $1$ | $ 3 \cdot 61 $ | 6.0.373837707.1 | $C_6$ (as 6T1) | $0$ | $-1$ | |
2.14884.3t2.a.a | $2$ | $ 2^{2} \cdot 61^{2}$ | 3.1.14884.1 | $S_3$ (as 3T2) | $1$ | $0$ | |
2.133956.6t3.b.a | $2$ | $ 2^{2} \cdot 3^{2} \cdot 61^{2}$ | 6.2.23925613248.4 | $D_{6}$ (as 6T3) | $1$ | $0$ | |
* | 2.244.6t5.a.a | $2$ | $ 2^{2} \cdot 61 $ | 6.0.238144.2 | $S_3\times C_3$ (as 6T5) | $0$ | $0$ |
* | 2.244.6t5.a.b | $2$ | $ 2^{2} \cdot 61 $ | 6.0.238144.2 | $S_3\times C_3$ (as 6T5) | $0$ | $0$ |
* | 2.2196.12t18.b.a | $2$ | $ 2^{2} \cdot 3^{2} \cdot 61 $ | 12.0.41343459692544.1 | $C_6\times S_3$ (as 12T18) | $0$ | $0$ |
* | 2.2196.12t18.b.b | $2$ | $ 2^{2} \cdot 3^{2} \cdot 61 $ | 12.0.41343459692544.1 | $C_6\times S_3$ (as 12T18) | $0$ | $0$ |