Normalized defining polynomial
\( x^{12} - 2x^{11} + 2x^{10} - x^{9} + 5x^{8} + 2x^{7} - 3x^{6} + 2x^{5} + 5x^{4} + 9x^{3} + 7x^{2} + 3x + 1 \)
Invariants
Degree: | $12$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[0, 6]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(1200500000000\) \(\medspace = 2^{8}\cdot 5^{9}\cdot 7^{4}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(10.15\) | 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: | $2^{2/3}5^{3/4}7^{1/2}\approx 14.043106421634382$ | ||
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: | \(\Q(\sqrt{5}) \) | ||
$\card{ \Aut(K/\Q) }$: | $4$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{31}a^{11}+\frac{2}{31}a^{10}+\frac{10}{31}a^{9}+\frac{8}{31}a^{8}+\frac{6}{31}a^{7}-\frac{5}{31}a^{6}+\frac{8}{31}a^{5}+\frac{3}{31}a^{4}-\frac{14}{31}a^{3}+\frac{15}{31}a^{2}+\frac{5}{31}a-\frac{8}{31}$
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)
| |
Torsion generator: | \( \frac{300}{31} a^{11} - \frac{826}{31} a^{10} + \frac{1233}{31} a^{9} - \frac{1258}{31} a^{8} + \frac{2482}{31} a^{7} - \frac{1283}{31} a^{6} + \frac{106}{31} a^{5} + \frac{528}{31} a^{4} + \frac{1039}{31} a^{3} + \frac{1989}{31} a^{2} + \frac{663}{31} a + \frac{452}{31} \) (order $10$) | 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: | $\frac{154}{31}a^{11}-\frac{405}{31}a^{10}+\frac{579}{31}a^{9}-\frac{566}{31}a^{8}+\frac{1203}{31}a^{7}-\frac{522}{31}a^{6}-\frac{8}{31}a^{5}+\frac{245}{31}a^{4}+\frac{603}{31}a^{3}+\frac{1101}{31}a^{2}+\frac{429}{31}a+\frac{256}{31}$, $\frac{300}{31}a^{11}-\frac{826}{31}a^{10}+\frac{1233}{31}a^{9}-\frac{1258}{31}a^{8}+\frac{2482}{31}a^{7}-\frac{1283}{31}a^{6}+\frac{106}{31}a^{5}+\frac{528}{31}a^{4}+\frac{1039}{31}a^{3}+\frac{1989}{31}a^{2}+\frac{663}{31}a+\frac{421}{31}$, $\frac{200}{31}a^{11}-\frac{561}{31}a^{10}+\frac{822}{31}a^{9}-\frac{787}{31}a^{8}+\frac{1541}{31}a^{7}-\frac{783}{31}a^{6}-\frac{136}{31}a^{5}+\frac{538}{31}a^{4}+\frac{672}{31}a^{3}+\frac{1140}{31}a^{2}+\frac{318}{31}a+\frac{167}{31}$, $\frac{96}{31}a^{11}-\frac{273}{31}a^{10}+\frac{402}{31}a^{9}-\frac{379}{31}a^{8}+\frac{731}{31}a^{7}-\frac{387}{31}a^{6}-\frac{69}{31}a^{5}+\frac{288}{31}a^{4}+\frac{330}{31}a^{3}+\frac{479}{31}a^{2}+\frac{108}{31}a+\frac{69}{31}$, $\frac{123}{31}a^{11}-\frac{374}{31}a^{10}+\frac{579}{31}a^{9}-\frac{566}{31}a^{8}+\frac{986}{31}a^{7}-\frac{615}{31}a^{6}-\frac{70}{31}a^{5}+\frac{462}{31}a^{4}+\frac{293}{31}a^{3}+\frac{543}{31}a^{2}+\frac{88}{31}a+\frac{70}{31}$ | 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: | \( 44.4179208274 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 44.4179208274 \cdot 1}{10\cdot\sqrt{1200500000000}}\cr\approx \mathstrut & 0.249434403396 \end{aligned}\]
Galois group
$C_4\times S_3$ (as 12T11):
A solvable group of order 24 |
The 12 conjugacy class representatives for $S_3 \times C_4$ |
Character table for $S_3 \times C_4$ |
Intermediate fields
\(\Q(\sqrt{5}) \), 3.1.140.1, \(\Q(\zeta_{5})\), 6.2.98000.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Galois closure: | deg 24 |
Degree 12 sibling: | 12.4.58824500000000.1 |
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 | ${\href{/padicField/3.12.0.1}{12} }$ | R | R | ${\href{/padicField/11.3.0.1}{3} }^{4}$ | ${\href{/padicField/13.12.0.1}{12} }$ | ${\href{/padicField/17.12.0.1}{12} }$ | ${\href{/padicField/19.2.0.1}{2} }^{6}$ | ${\href{/padicField/23.4.0.1}{4} }^{3}$ | ${\href{/padicField/29.6.0.1}{6} }^{2}$ | ${\href{/padicField/31.2.0.1}{2} }^{4}{,}\,{\href{/padicField/31.1.0.1}{1} }^{4}$ | ${\href{/padicField/37.4.0.1}{4} }^{3}$ | ${\href{/padicField/41.2.0.1}{2} }^{4}{,}\,{\href{/padicField/41.1.0.1}{1} }^{4}$ | ${\href{/padicField/43.4.0.1}{4} }^{3}$ | ${\href{/padicField/47.12.0.1}{12} }$ | ${\href{/padicField/53.4.0.1}{4} }^{3}$ | ${\href{/padicField/59.2.0.1}{2} }^{6}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.12.8.1 | $x^{12} + 11 x^{9} + 3 x^{8} - 9 x^{6} - 90 x^{5} + 3 x^{4} - 27 x^{3} + 135 x^{2} + 27 x + 55$ | $3$ | $4$ | $8$ | $C_3 : C_4$ | $[\ ]_{3}^{4}$ |
\(5\) | 5.4.3.2 | $x^{4} + 5$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ |
5.8.6.1 | $x^{8} + 16 x^{7} + 104 x^{6} + 352 x^{5} + 674 x^{4} + 784 x^{3} + 776 x^{2} + 928 x + 721$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ | |
\(7\) | 7.4.0.1 | $x^{4} + 5 x^{2} + 4 x + 3$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |
7.8.4.1 | $x^{8} + 38 x^{6} + 8 x^{5} + 395 x^{4} - 72 x^{3} + 1026 x^{2} - 872 x + 401$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
1.35.2t1.a.a | $1$ | $ 5 \cdot 7 $ | \(\Q(\sqrt{-35}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
* | 1.5.2t1.a.a | $1$ | $ 5 $ | \(\Q(\sqrt{5}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
1.7.2t1.a.a | $1$ | $ 7 $ | \(\Q(\sqrt{-7}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
1.35.4t1.a.a | $1$ | $ 5 \cdot 7 $ | 4.4.6125.1 | $C_4$ (as 4T1) | $0$ | $1$ | |
* | 1.5.4t1.a.a | $1$ | $ 5 $ | \(\Q(\zeta_{5})\) | $C_4$ (as 4T1) | $0$ | $-1$ |
1.35.4t1.a.b | $1$ | $ 5 \cdot 7 $ | 4.4.6125.1 | $C_4$ (as 4T1) | $0$ | $1$ | |
* | 1.5.4t1.a.b | $1$ | $ 5 $ | \(\Q(\zeta_{5})\) | $C_4$ (as 4T1) | $0$ | $-1$ |
* | 2.140.3t2.a.a | $2$ | $ 2^{2} \cdot 5 \cdot 7 $ | 3.1.140.1 | $S_3$ (as 3T2) | $1$ | $0$ |
* | 2.140.6t3.b.a | $2$ | $ 2^{2} \cdot 5 \cdot 7 $ | 6.2.98000.1 | $D_{6}$ (as 6T3) | $1$ | $0$ |
* | 2.700.12t11.a.a | $2$ | $ 2^{2} \cdot 5^{2} \cdot 7 $ | 12.0.1200500000000.1 | $S_3 \times C_4$ (as 12T11) | $0$ | $0$ |
* | 2.700.12t11.a.b | $2$ | $ 2^{2} \cdot 5^{2} \cdot 7 $ | 12.0.1200500000000.1 | $S_3 \times C_4$ (as 12T11) | $0$ | $0$ |