Normalized defining polynomial
\( x^{12} - 3x^{10} - 8x^{9} + x^{8} + 24x^{7} + 14x^{6} - 8x^{5} + 5x^{4} + 8x^{3} - 4x^{2} - 2x + 1 \)
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: | \(1827904000000\) \(\medspace = 2^{12}\cdot 5^{6}\cdot 13^{4}\) | 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: | \(10.52\) | 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: | $2\cdot 5^{1/2}13^{2/3}\approx 24.725432631349385$ | ||
Ramified primes: | \(2\), \(5\), \(13\) | 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)
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This field is not Galois over $\Q$. | |||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{113129}a^{11}-\frac{14931}{113129}a^{10}-\frac{42501}{113129}a^{9}+\frac{41862}{113129}a^{8}-\frac{3796}{113129}a^{7}+\frac{471}{113129}a^{6}-\frac{18489}{113129}a^{5}+\frac{24491}{113129}a^{4}-\frac{42188}{113129}a^{3}+\frac{6764}{113129}a^{2}+\frac{30909}{113129}a-\frac{49090}{113129}$
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: | \( \frac{286}{3901} a^{11} - \frac{2572}{3901} a^{10} + \frac{230}{3901} a^{9} + \frac{4264}{3901} a^{8} + \frac{18327}{3901} a^{7} - \frac{1829}{3901} a^{6} - \frac{48811}{3901} a^{5} - \frac{17374}{3901} a^{4} + \frac{15629}{3901} a^{3} - \frac{23798}{3901} a^{2} - \frac{11395}{3901} a + \frac{7761}{3901} \) (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)
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Fundamental units: | $\frac{60598}{113129}a^{11}+\frac{17004}{113129}a^{10}-\frac{207042}{113129}a^{9}-\frac{503736}{113129}a^{8}-\frac{38751}{113129}a^{7}+\frac{1616956}{113129}a^{6}+\frac{1051355}{113129}a^{5}-\frac{599378}{113129}a^{4}+\frac{320105}{113129}a^{3}+\frac{584150}{113129}a^{2}-\frac{279529}{113129}a-\frac{141894}{113129}$, $\frac{1482}{113129}a^{11}-\frac{67587}{113129}a^{10}+\frac{26371}{113129}a^{9}+\frac{157921}{113129}a^{8}+\frac{483294}{113129}a^{7}-\frac{207010}{113129}a^{6}-\frac{1381028}{113129}a^{5}-\frac{471263}{113129}a^{4}+\frac{377108}{113129}a^{3}-\frac{383620}{113129}a^{2}-\frac{123236}{113129}a+\frac{216825}{113129}$, $\frac{27061}{113129}a^{11}-\frac{64132}{113129}a^{10}-\frac{50147}{113129}a^{9}-\frac{46224}{113129}a^{8}+\frac{450092}{113129}a^{7}+\frac{414670}{113129}a^{6}-\frac{979423}{113129}a^{5}-\frac{411247}{113129}a^{4}+\frac{500916}{113129}a^{3}-\frac{454634}{113129}a^{2}-\frac{273635}{113129}a+\frac{162486}{113129}$, $\frac{78448}{113129}a^{11}+\frac{30578}{113129}a^{10}-\frac{206818}{113129}a^{9}-\frac{710339}{113129}a^{8}-\frac{259338}{113129}a^{7}+\frac{1652760}{113129}a^{6}+\frac{1811901}{113129}a^{5}+\frac{565806}{113129}a^{4}+\frac{816574}{113129}a^{3}+\frac{499778}{113129}a^{2}-\frac{170883}{113129}a-\frac{101160}{113129}$, $\frac{1470}{113129}a^{11}-\frac{1544}{113129}a^{10}-\frac{29262}{113129}a^{9}-\frac{5036}{113129}a^{8}+\frac{76330}{113129}a^{7}+\frac{239854}{113129}a^{6}-\frac{27870}{113129}a^{5}-\frac{538897}{113129}a^{4}-\frac{361055}{113129}a^{3}-\frac{12272}{113129}a^{2}-\frac{154757}{113129}a-\frac{99127}{113129}$ | 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: | \( 18.262093981439413 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 18.262093981439413 \cdot 1}{4\cdot\sqrt{1827904000000}}\cr\approx \mathstrut & 0.207774908944526 \end{aligned}\]
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
\(\Q(\sqrt{-5}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-1}) \), \(\Q(i, \sqrt{5})\), 6.0.10816.1 |
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: | 18.6.186384680979848000000000.1, 18.0.11928619582710272000000000.1 |
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 | ${\href{/padicField/3.6.0.1}{6} }^{2}$ | R | ${\href{/padicField/7.6.0.1}{6} }^{2}$ | ${\href{/padicField/11.6.0.1}{6} }^{2}$ | R | ${\href{/padicField/17.6.0.1}{6} }{,}\,{\href{/padicField/17.2.0.1}{2} }^{3}$ | ${\href{/padicField/19.6.0.1}{6} }^{2}$ | ${\href{/padicField/23.6.0.1}{6} }^{2}$ | ${\href{/padicField/29.3.0.1}{3} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }^{6}$ | ${\href{/padicField/31.2.0.1}{2} }^{6}$ | ${\href{/padicField/37.6.0.1}{6} }{,}\,{\href{/padicField/37.2.0.1}{2} }^{3}$ | ${\href{/padicField/41.3.0.1}{3} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }^{6}$ | ${\href{/padicField/43.6.0.1}{6} }^{2}$ | ${\href{/padicField/47.2.0.1}{2} }^{6}$ | ${\href{/padicField/53.6.0.1}{6} }^{2}$ | ${\href{/padicField/59.6.0.1}{6} }^{2}$ |
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 | $x^{12} + 12 x^{11} + 98 x^{10} + 542 x^{9} + 2359 x^{8} + 7956 x^{7} + 21831 x^{6} + 47308 x^{5} + 82476 x^{4} + 109442 x^{3} + 112071 x^{2} + 76900 x + 33205$ | $2$ | $6$ | $12$ | $C_6\times C_2$ | $[2]^{6}$ |
\(5\) | 5.6.3.1 | $x^{6} + 60 x^{5} + 1221 x^{4} + 8846 x^{3} + 9864 x^{2} + 29208 x + 29309$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ |
5.6.3.1 | $x^{6} + 60 x^{5} + 1221 x^{4} + 8846 x^{3} + 9864 x^{2} + 29208 x + 29309$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
\(13\) | 13.6.0.1 | $x^{6} + 10 x^{3} + 11 x^{2} + 11 x + 2$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ |
13.6.4.1 | $x^{6} + 130 x^{3} - 1521$ | $3$ | $2$ | $4$ | $C_6$ | $[\ ]_{3}^{2}$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
* | 1.20.2t1.a.a | $1$ | $ 2^{2} \cdot 5 $ | \(\Q(\sqrt{-5}) \) | $C_2$ (as 2T1) | $1$ | $-1$ |
* | 1.4.2t1.a.a | $1$ | $ 2^{2}$ | \(\Q(\sqrt{-1}) \) | $C_2$ (as 2T1) | $1$ | $-1$ |
* | 1.5.2t1.a.a | $1$ | $ 5 $ | \(\Q(\sqrt{5}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
1.65.6t1.b.a | $1$ | $ 5 \cdot 13 $ | 6.6.3570125.1 | $C_6$ (as 6T1) | $0$ | $1$ | |
1.260.6t1.a.a | $1$ | $ 2^{2} \cdot 5 \cdot 13 $ | 6.0.228488000.1 | $C_6$ (as 6T1) | $0$ | $-1$ | |
1.260.6t1.a.b | $1$ | $ 2^{2} \cdot 5 \cdot 13 $ | 6.0.228488000.1 | $C_6$ (as 6T1) | $0$ | $-1$ | |
1.52.6t1.b.a | $1$ | $ 2^{2} \cdot 13 $ | 6.0.1827904.1 | $C_6$ (as 6T1) | $0$ | $-1$ | |
1.13.3t1.a.a | $1$ | $ 13 $ | 3.3.169.1 | $C_3$ (as 3T1) | $0$ | $1$ | |
1.65.6t1.b.b | $1$ | $ 5 \cdot 13 $ | 6.6.3570125.1 | $C_6$ (as 6T1) | $0$ | $1$ | |
1.13.3t1.a.b | $1$ | $ 13 $ | 3.3.169.1 | $C_3$ (as 3T1) | $0$ | $1$ | |
1.52.6t1.b.b | $1$ | $ 2^{2} \cdot 13 $ | 6.0.1827904.1 | $C_6$ (as 6T1) | $0$ | $-1$ | |
2.676.3t2.b.a | $2$ | $ 2^{2} \cdot 13^{2}$ | 3.1.676.1 | $S_3$ (as 3T2) | $1$ | $0$ | |
2.16900.6t3.a.a | $2$ | $ 2^{2} \cdot 5^{2} \cdot 13^{2}$ | 6.0.228488000.2 | $D_{6}$ (as 6T3) | $1$ | $0$ | |
* | 2.1300.12t18.c.a | $2$ | $ 2^{2} \cdot 5^{2} \cdot 13 $ | 12.0.1827904000000.1 | $C_6\times S_3$ (as 12T18) | $0$ | $0$ |
* | 2.52.6t5.b.a | $2$ | $ 2^{2} \cdot 13 $ | 6.0.10816.1 | $S_3\times C_3$ (as 6T5) | $0$ | $0$ |
* | 2.1300.12t18.c.b | $2$ | $ 2^{2} \cdot 5^{2} \cdot 13 $ | 12.0.1827904000000.1 | $C_6\times S_3$ (as 12T18) | $0$ | $0$ |
* | 2.52.6t5.b.b | $2$ | $ 2^{2} \cdot 13 $ | 6.0.10816.1 | $S_3\times C_3$ (as 6T5) | $0$ | $0$ |