Normalized defining polynomial
\( x^{12} - 2 x^{11} - 2 x^{10} + 2 x^{9} + 64 x^{8} - 112 x^{7} + 78 x^{6} + 112 x^{5} - 104 x^{4} + \cdots + 36 \)
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: | \(289254654976000000\) \(\medspace = 2^{16}\cdot 5^{6}\cdot 7^{10}\) | 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: | \(28.52\) | 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))
| |
Galois root discriminant: | $2^{4/3}5^{1/2}7^{5/6}\approx 28.517187846962024$ | ||
Ramified primes: | \(2\), \(5\), \(7\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q\) | ||
$\card{ \Aut(K/\Q) }$: | $6$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2}a^{6}$, $\frac{1}{2}a^{7}$, $\frac{1}{6}a^{8}+\frac{1}{6}a^{7}+\frac{1}{6}a^{6}-\frac{1}{3}a^{5}-\frac{1}{3}a^{3}+\frac{1}{3}a^{2}-\frac{1}{3}a$, $\frac{1}{18}a^{9}-\frac{1}{18}a^{8}+\frac{1}{9}a^{7}+\frac{1}{9}a^{6}-\frac{4}{9}a^{5}+\frac{2}{9}a^{4}-\frac{1}{3}a^{3}-\frac{1}{3}a^{2}-\frac{1}{9}a+\frac{1}{3}$, $\frac{1}{54}a^{10}+\frac{1}{54}a^{8}+\frac{13}{54}a^{7}+\frac{1}{18}a^{6}-\frac{11}{27}a^{5}-\frac{10}{27}a^{4}-\frac{2}{9}a^{3}+\frac{5}{27}a^{2}-\frac{7}{27}a+\frac{4}{9}$, $\frac{1}{1296584334}a^{11}+\frac{2878487}{432194778}a^{10}+\frac{2408324}{648292167}a^{9}-\frac{3825958}{648292167}a^{8}-\frac{655103}{8003607}a^{7}+\frac{6649063}{648292167}a^{6}+\frac{301101593}{648292167}a^{5}-\frac{71181247}{216097389}a^{4}-\frac{277490416}{648292167}a^{3}+\frac{273914105}{648292167}a^{2}+\frac{21021248}{216097389}a+\frac{23901557}{72032463}$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $3$ |
Class group and class number
$C_{3}\times C_{6}$, which has order $18$
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)
| |
Fundamental units: | $\frac{31646789}{432194778}a^{11}-\frac{4916261}{72032463}a^{10}-\frac{50827910}{216097389}a^{9}-\frac{37074499}{432194778}a^{8}+\frac{334344704}{72032463}a^{7}-\frac{697539733}{216097389}a^{6}+\frac{263446054}{216097389}a^{5}+\frac{251947201}{24010821}a^{4}+\frac{675482782}{216097389}a^{3}-\frac{3870062171}{216097389}a^{2}+\frac{1740370}{2667869}a+\frac{4939901}{24010821}$, $\frac{10757083}{1296584334}a^{11}-\frac{7087183}{432194778}a^{10}-\frac{6191521}{648292167}a^{9}+\frac{8158493}{648292167}a^{8}+\frac{36195718}{72032463}a^{7}-\frac{615799862}{648292167}a^{6}+\frac{721031726}{648292167}a^{5}+\frac{186625076}{216097389}a^{4}-\frac{635231770}{648292167}a^{3}-\frac{1342272103}{648292167}a^{2}+\frac{856683308}{216097389}a-\frac{168290281}{72032463}$, $\frac{13849777}{216097389}a^{11}-\frac{2095285}{24010821}a^{10}-\frac{78842491}{432194778}a^{9}+\frac{5546087}{432194778}a^{8}+\frac{592606409}{144064926}a^{7}-\frac{987793204}{216097389}a^{6}+\frac{460976497}{216097389}a^{5}+\frac{619093421}{72032463}a^{4}-\frac{128466332}{216097389}a^{3}-\frac{3800420420}{216097389}a^{2}+\frac{590908562}{72032463}a-\frac{5048331}{2667869}$, $\frac{28640135}{432194778}a^{11}-\frac{4613689}{72032463}a^{10}-\frac{88999567}{432194778}a^{9}-\frac{29676787}{432194778}a^{8}+\frac{201309431}{48021642}a^{7}-\frac{671818693}{216097389}a^{6}+\frac{327403261}{216097389}a^{5}+\frac{690564004}{72032463}a^{4}+\frac{637245055}{216097389}a^{3}-\frac{3563882414}{216097389}a^{2}+\frac{317150011}{72032463}a-\frac{61434959}{24010821}$, $a-1$ | 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: | \( 3296.9133952651964 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 3296.9133952651964 \cdot 18}{2\cdot\sqrt{289254654976000000}}\cr\approx \mathstrut & 3.39460370003625 \end{aligned}\]
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
\(\Q(\sqrt{-5}) \), \(\Q(\sqrt{-35}) \), \(\Q(\sqrt{7}) \), \(\Q(\sqrt{-5}, \sqrt{7})\), 6.0.33614000.1 |
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: | 18.6.1244544764462497792000000.1, 18.0.22224013651116032000000000.1 |
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 | ${\href{/padicField/3.3.0.1}{3} }^{2}{,}\,{\href{/padicField/3.1.0.1}{1} }^{6}$ | R | R | ${\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.2.0.1}{2} }^{3}$ | ${\href{/padicField/13.6.0.1}{6} }^{2}$ | ${\href{/padicField/17.6.0.1}{6} }{,}\,{\href{/padicField/17.2.0.1}{2} }^{3}$ | ${\href{/padicField/19.6.0.1}{6} }^{2}$ | ${\href{/padicField/23.6.0.1}{6} }^{2}$ | ${\href{/padicField/29.3.0.1}{3} }^{4}$ | ${\href{/padicField/31.6.0.1}{6} }^{2}$ | ${\href{/padicField/37.6.0.1}{6} }^{2}$ | ${\href{/padicField/41.2.0.1}{2} }^{6}$ | ${\href{/padicField/43.2.0.1}{2} }^{6}$ | ${\href{/padicField/47.3.0.1}{3} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }^{6}$ | ${\href{/padicField/53.6.0.1}{6} }^{2}$ | ${\href{/padicField/59.6.0.1}{6} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.12.16.3 | $x^{12} + 4 x^{11} + 8 x^{10} + 10 x^{9} + 8 x^{8} + 4 x^{7} + 2 x^{6} - 4 x^{5} - 4 x^{4} + 4 x^{3} + 4$ | $6$ | $2$ | $16$ | $C_6\times S_3$ | $[2]_{3}^{6}$ |
\(5\) | 5.12.6.1 | $x^{12} + 120 x^{11} + 6032 x^{10} + 163208 x^{9} + 2529528 x^{8} + 21853448 x^{7} + 92223962 x^{6} + 138649448 x^{5} + 223472880 x^{4} + 401794296 x^{3} + 295909124 x^{2} + 118616440 x + 126881009$ | $2$ | $6$ | $6$ | $C_6\times C_2$ | $[\ ]_{2}^{6}$ |
\(7\) | 7.6.5.2 | $x^{6} + 42$ | $6$ | $1$ | $5$ | $C_6$ | $[\ ]_{6}$ |
7.6.5.2 | $x^{6} + 42$ | $6$ | $1$ | $5$ | $C_6$ | $[\ ]_{6}$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
* | 1.28.2t1.a.a | $1$ | $ 2^{2} \cdot 7 $ | \(\Q(\sqrt{7}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
* | 1.35.2t1.a.a | $1$ | $ 5 \cdot 7 $ | \(\Q(\sqrt{-35}) \) | $C_2$ (as 2T1) | $1$ | $-1$ |
* | 1.20.2t1.a.a | $1$ | $ 2^{2} \cdot 5 $ | \(\Q(\sqrt{-5}) \) | $C_2$ (as 2T1) | $1$ | $-1$ |
1.35.6t1.a.a | $1$ | $ 5 \cdot 7 $ | 6.0.2100875.1 | $C_6$ (as 6T1) | $0$ | $-1$ | |
1.140.6t1.b.a | $1$ | $ 2^{2} \cdot 5 \cdot 7 $ | 6.0.19208000.1 | $C_6$ (as 6T1) | $0$ | $-1$ | |
1.7.3t1.a.a | $1$ | $ 7 $ | \(\Q(\zeta_{7})^+\) | $C_3$ (as 3T1) | $0$ | $1$ | |
1.7.3t1.a.b | $1$ | $ 7 $ | \(\Q(\zeta_{7})^+\) | $C_3$ (as 3T1) | $0$ | $1$ | |
1.28.6t1.b.a | $1$ | $ 2^{2} \cdot 7 $ | \(\Q(\zeta_{28})^+\) | $C_6$ (as 6T1) | $0$ | $1$ | |
1.35.6t1.a.b | $1$ | $ 5 \cdot 7 $ | 6.0.2100875.1 | $C_6$ (as 6T1) | $0$ | $-1$ | |
1.28.6t1.b.b | $1$ | $ 2^{2} \cdot 7 $ | \(\Q(\zeta_{28})^+\) | $C_6$ (as 6T1) | $0$ | $1$ | |
1.140.6t1.b.b | $1$ | $ 2^{2} \cdot 5 \cdot 7 $ | 6.0.19208000.1 | $C_6$ (as 6T1) | $0$ | $-1$ | |
2.140.3t2.a.a | $2$ | $ 2^{2} \cdot 5 \cdot 7 $ | 3.1.140.1 | $S_3$ (as 3T2) | $1$ | $0$ | |
2.560.6t3.a.a | $2$ | $ 2^{4} \cdot 5 \cdot 7 $ | 6.2.2195200.2 | $D_{6}$ (as 6T3) | $1$ | $0$ | |
* | 2.980.6t5.a.a | $2$ | $ 2^{2} \cdot 5 \cdot 7^{2}$ | 6.0.33614000.1 | $S_3\times C_3$ (as 6T5) | $0$ | $0$ |
* | 2.980.6t5.a.b | $2$ | $ 2^{2} \cdot 5 \cdot 7^{2}$ | 6.0.33614000.1 | $S_3\times C_3$ (as 6T5) | $0$ | $0$ |
* | 2.3920.12t18.a.a | $2$ | $ 2^{4} \cdot 5 \cdot 7^{2}$ | 12.0.289254654976000000.1 | $C_6\times S_3$ (as 12T18) | $0$ | $0$ |
* | 2.3920.12t18.a.b | $2$ | $ 2^{4} \cdot 5 \cdot 7^{2}$ | 12.0.289254654976000000.1 | $C_6\times S_3$ (as 12T18) | $0$ | $0$ |