Normalized defining polynomial
\( x^{12} + 3x^{10} - 10x^{9} + 4x^{8} - 30x^{7} + 41x^{6} - 40x^{5} + 92x^{4} - 80x^{3} + 128x^{2} - 64x + 64 \)
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: | \(13763268972544\) \(\medspace = 2^{12}\cdot 7^{6}\cdot 13^{4}\) | 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: | \(12.44\) | 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: | $2\cdot 7^{1/2}13^{2/3}\approx 29.255526423743525$ | ||
Ramified primes: | \(2\), \(7\), \(13\) | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a$, $\frac{1}{4}a^{8}-\frac{1}{4}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}+\frac{1}{4}a^{2}$, $\frac{1}{8}a^{9}-\frac{1}{8}a^{7}-\frac{1}{4}a^{6}+\frac{1}{4}a^{4}+\frac{1}{8}a^{3}$, $\frac{1}{16}a^{10}-\frac{1}{16}a^{8}-\frac{1}{8}a^{7}-\frac{1}{2}a^{6}+\frac{1}{8}a^{5}-\frac{7}{16}a^{4}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{5689568}a^{11}+\frac{3193}{2844784}a^{10}+\frac{242827}{5689568}a^{9}+\frac{71483}{1422392}a^{8}+\frac{40203}{711196}a^{7}-\frac{732803}{2844784}a^{6}-\frac{20515}{5689568}a^{5}-\frac{785579}{2844784}a^{4}+\frac{19585}{49048}a^{3}+\frac{318033}{711196}a^{2}-\frac{108511}{355598}a+\frac{54626}{177799}$
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: | \( \frac{733}{196192} a^{11} - \frac{1573}{98096} a^{10} - \frac{3001}{196192} a^{9} - \frac{2309}{24524} a^{8} + \frac{3213}{24524} a^{7} + \frac{4573}{98096} a^{6} + \frac{118337}{196192} a^{5} - \frac{42673}{98096} a^{4} + \frac{11183}{49048} a^{3} - \frac{18805}{12262} a^{2} + \frac{5581}{6131} a - \frac{6834}{6131} \) (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: | $\frac{13479}{5689568}a^{11}+\frac{11089}{2844784}a^{10}+\frac{141141}{5689568}a^{9}+\frac{3322}{177799}a^{8}-\frac{35115}{711196}a^{7}-\frac{361589}{2844784}a^{6}-\frac{2000029}{5689568}a^{5}+\frac{533501}{2844784}a^{4}-\frac{2383}{49048}a^{3}+\frac{744113}{711196}a^{2}-\frac{111497}{177799}a+\frac{215994}{177799}$, $\frac{1113}{355598}a^{11}-\frac{2171}{177799}a^{10}+\frac{11971}{355598}a^{9}-\frac{8947}{177799}a^{8}+\frac{58125}{355598}a^{7}-\frac{45726}{177799}a^{6}+\frac{51438}{177799}a^{5}-\frac{111744}{177799}a^{4}+\frac{4800}{6131}a^{3}-\frac{108320}{177799}a^{2}+\frac{133443}{355598}a-\frac{136320}{177799}$, $\frac{5773}{1422392}a^{11}-\frac{53829}{2844784}a^{10}+\frac{72955}{1422392}a^{9}-\frac{178759}{2844784}a^{8}+\frac{335993}{1422392}a^{7}-\frac{277911}{711196}a^{6}+\frac{128609}{355598}a^{5}-\frac{3492077}{2844784}a^{4}+\frac{7348}{6131}a^{3}-\frac{125764}{177799}a^{2}+\frac{259346}{177799}a-\frac{60313}{177799}$, $\frac{421}{355598}a^{11}-\frac{5631}{2844784}a^{10}-\frac{2129}{177799}a^{9}+\frac{237391}{2844784}a^{8}-\frac{138737}{1422392}a^{7}+\frac{120203}{355598}a^{6}-\frac{1298847}{1422392}a^{5}+\frac{2305081}{2844784}a^{4}-\frac{22033}{12262}a^{3}+\frac{962963}{355598}a^{2}-\frac{265902}{177799}a+\frac{272404}{177799}$, $\frac{48091}{1422392}a^{11}+\frac{50253}{1422392}a^{10}+\frac{16592}{177799}a^{9}-\frac{396647}{1422392}a^{8}-\frac{404221}{1422392}a^{7}-\frac{289139}{355598}a^{6}+\frac{908781}{1422392}a^{5}+\frac{859451}{1422392}a^{4}+\frac{78191}{49048}a^{3}-\frac{23351}{355598}a^{2}-\frac{3702}{177799}a+\frac{510562}{177799}$ | 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: | \( 62.145765179685824 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 62.145765179685824 \cdot 1}{4\cdot\sqrt{13763268972544}}\cr\approx \mathstrut & 0.257673620638546 \end{aligned}\]
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
\(\Q(\sqrt{-7}) \), \(\Q(\sqrt{7}) \), \(\Q(\sqrt{-1}) \), \(\Q(i, \sqrt{7})\), 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: | 18.6.246457767266915487309365248.1, 18.0.3850902613545554489208832.1 |
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 | ${\href{/padicField/3.6.0.1}{6} }^{2}$ | ${\href{/padicField/5.6.0.1}{6} }^{2}$ | R | ${\href{/padicField/11.6.0.1}{6} }^{2}$ | R | ${\href{/padicField/17.6.0.1}{6} }{,}\,{\href{/padicField/17.2.0.1}{2} }^{3}$ | ${\href{/padicField/19.6.0.1}{6} }^{2}$ | ${\href{/padicField/23.6.0.1}{6} }^{2}$ | ${\href{/padicField/29.3.0.1}{3} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }^{6}$ | ${\href{/padicField/31.2.0.1}{2} }^{6}$ | ${\href{/padicField/37.3.0.1}{3} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }^{6}$ | ${\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.2.0.1}{2} }^{3}$ | ${\href{/padicField/43.6.0.1}{6} }^{2}$ | ${\href{/padicField/47.2.0.1}{2} }^{6}$ | ${\href{/padicField/53.3.0.1}{3} }^{4}$ | ${\href{/padicField/59.6.0.1}{6} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.6.6.3 | $x^{6} + 6 x^{5} + 20 x^{4} + 42 x^{3} + 55 x^{2} + 36 x + 9$ | $2$ | $3$ | $6$ | $C_6$ | $[2]^{3}$ |
2.6.6.3 | $x^{6} + 6 x^{5} + 20 x^{4} + 42 x^{3} + 55 x^{2} + 36 x + 9$ | $2$ | $3$ | $6$ | $C_6$ | $[2]^{3}$ | |
\(7\) | 7.12.6.1 | $x^{12} + 44 x^{10} + 10 x^{9} + 786 x^{8} + 22 x^{7} + 6899 x^{6} - 3434 x^{5} + 31050 x^{4} - 28440 x^{3} + 84557 x^{2} - 48082 x + 107648$ | $2$ | $6$ | $6$ | $C_6\times C_2$ | $[\ ]_{2}^{6}$ |
\(13\) | 13.6.4.1 | $x^{6} + 130 x^{3} - 1521$ | $3$ | $2$ | $4$ | $C_6$ | $[\ ]_{3}^{2}$ |
13.6.0.1 | $x^{6} + 10 x^{3} + 11 x^{2} + 11 x + 2$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
* | 1.7.2t1.a.a | $1$ | $ 7 $ | \(\Q(\sqrt{-7}) \) | $C_2$ (as 2T1) | $1$ | $-1$ |
* | 1.4.2t1.a.a | $1$ | $ 2^{2}$ | \(\Q(\sqrt{-1}) \) | $C_2$ (as 2T1) | $1$ | $-1$ |
* | 1.28.2t1.a.a | $1$ | $ 2^{2} \cdot 7 $ | \(\Q(\sqrt{7}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
1.91.6t1.g.a | $1$ | $ 7 \cdot 13 $ | 6.0.9796423.1 | $C_6$ (as 6T1) | $0$ | $-1$ | |
1.364.6t1.h.a | $1$ | $ 2^{2} \cdot 7 \cdot 13 $ | 6.6.626971072.1 | $C_6$ (as 6T1) | $0$ | $1$ | |
1.52.6t1.b.a | $1$ | $ 2^{2} \cdot 13 $ | 6.0.1827904.1 | $C_6$ (as 6T1) | $0$ | $-1$ | |
1.52.6t1.b.b | $1$ | $ 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) | $0$ | $1$ | |
1.13.3t1.a.b | $1$ | $ 13 $ | 3.3.169.1 | $C_3$ (as 3T1) | $0$ | $1$ | |
1.91.6t1.g.b | $1$ | $ 7 \cdot 13 $ | 6.0.9796423.1 | $C_6$ (as 6T1) | $0$ | $-1$ | |
1.364.6t1.h.b | $1$ | $ 2^{2} \cdot 7 \cdot 13 $ | 6.6.626971072.1 | $C_6$ (as 6T1) | $0$ | $1$ | |
2.676.3t2.b.a | $2$ | $ 2^{2} \cdot 13^{2}$ | 3.1.676.1 | $S_3$ (as 3T2) | $1$ | $0$ | |
2.33124.6t3.b.a | $2$ | $ 2^{2} \cdot 7^{2} \cdot 13^{2}$ | 6.2.626971072.1 | $D_{6}$ (as 6T3) | $1$ | $0$ | |
* | 2.52.6t5.b.a | $2$ | $ 2^{2} \cdot 13 $ | 6.0.10816.1 | $S_3\times C_3$ (as 6T5) | $0$ | $0$ |
* | 2.2548.12t18.f.a | $2$ | $ 2^{2} \cdot 7^{2} \cdot 13 $ | 12.0.13763268972544.2 | $C_6\times S_3$ (as 12T18) | $0$ | $0$ |
* | 2.2548.12t18.f.b | $2$ | $ 2^{2} \cdot 7^{2} \cdot 13 $ | 12.0.13763268972544.2 | $C_6\times S_3$ (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) | $0$ | $0$ |