Normalized defining polynomial
\( x^{12} - x^{11} + 3x^{10} - 4x^{9} + 9x^{8} + 2x^{7} + 12x^{6} + x^{5} + 25x^{4} - 11x^{3} + 5x^{2} - 2x + 1 \)
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: | \(11259376953125\) \(\medspace = 5^{9}\cdot 7^{8}\) | 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: | \(12.24\) | 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: | $5^{3/4}7^{2/3}\approx 12.235626082613623$ | ||
Ramified primes: | \(5\), \(7\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
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Discriminant root field: | \(\Q(\sqrt{5}) \) | ||
$\card{ \Gal(K/\Q) }$: | $12$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
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This field is Galois and abelian over $\Q$. | |||
Conductor: | \(35=5\cdot 7\) | ||
Dirichlet character group: | $\lbrace$$\chi_{35}(32,·)$, $\chi_{35}(1,·)$, $\chi_{35}(2,·)$, $\chi_{35}(4,·)$, $\chi_{35}(8,·)$, $\chi_{35}(9,·)$, $\chi_{35}(11,·)$, $\chi_{35}(16,·)$, $\chi_{35}(18,·)$, $\chi_{35}(22,·)$, $\chi_{35}(23,·)$, $\chi_{35}(29,·)$$\rbrace$ | ||
This is a CM field. | |||
Reflex fields: | \(\Q(\zeta_{5})\)$^{2}$, 12.0.11259376953125.1$^{30}$ |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{181}a^{9}+\frac{43}{181}a^{8}+\frac{39}{181}a^{7}+\frac{48}{181}a^{6}+\frac{73}{181}a^{5}+\frac{39}{181}a^{4}+\frac{48}{181}a^{3}+\frac{73}{181}a^{2}+\frac{62}{181}a-\frac{49}{181}$, $\frac{1}{181}a^{10}-\frac{23}{181}a^{5}-\frac{65}{181}$, $\frac{1}{181}a^{11}-\frac{23}{181}a^{6}-\frac{65}{181}a$
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: | \( \frac{14}{181} a^{11} - \frac{42}{181} a^{10} + \frac{56}{181} a^{9} - \frac{126}{181} a^{8} + \frac{193}{181} a^{7} - \frac{168}{181} a^{6} - \frac{14}{181} a^{5} - \frac{350}{181} a^{4} + \frac{154}{181} a^{3} - \frac{980}{181} a^{2} + \frac{28}{181} a - \frac{14}{181} \) (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)
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Fundamental units: | $\frac{9}{181}a^{11}+\frac{155}{181}a^{6}-\frac{404}{181}a+1$, $\frac{28}{181}a^{11}-\frac{84}{181}a^{10}+\frac{112}{181}a^{9}-\frac{252}{181}a^{8}+\frac{386}{181}a^{7}-\frac{336}{181}a^{6}-\frac{28}{181}a^{5}-\frac{700}{181}a^{4}+\frac{308}{181}a^{3}-\frac{1779}{181}a^{2}+\frac{56}{181}a-\frac{28}{181}$, $\frac{9}{181}a^{11}-\frac{27}{181}a^{10}+\frac{36}{181}a^{9}-\frac{81}{181}a^{8}+\frac{137}{181}a^{7}-\frac{108}{181}a^{6}-\frac{9}{181}a^{5}-\frac{225}{181}a^{4}+\frac{99}{181}a^{3}-\frac{449}{181}a^{2}+\frac{18}{181}a-\frac{9}{181}$, $\frac{88}{181}a^{11}-\frac{74}{181}a^{10}+\frac{241}{181}a^{9}-\frac{316}{181}a^{8}+\frac{711}{181}a^{7}+\frac{313}{181}a^{6}+\frac{1014}{181}a^{5}+\frac{168}{181}a^{4}+\frac{1975}{181}a^{3}-\frac{869}{181}a^{2}-\frac{9}{181}a-\frac{302}{181}$, $\frac{88}{181}a^{11}-\frac{70}{181}a^{10}+\frac{241}{181}a^{9}-\frac{316}{181}a^{8}+\frac{711}{181}a^{7}+\frac{313}{181}a^{6}+\frac{1103}{181}a^{5}+\frac{168}{181}a^{4}+\frac{1975}{181}a^{3}-\frac{869}{181}a^{2}-\frac{9}{181}a-\frac{381}{181}$ | 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: | \( 104.882003477 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 104.882003477 \cdot 1}{10\cdot\sqrt{11259376953125}}\cr\approx \mathstrut & 0.192319360106 \end{aligned}\]
Galois group
A cyclic group of order 12 |
The 12 conjugacy class representatives for $C_{12}$ |
Character table for $C_{12}$ |
Intermediate fields
\(\Q(\sqrt{5}) \), \(\Q(\zeta_{7})^+\), \(\Q(\zeta_{5})\), 6.6.300125.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | ${\href{/padicField/2.12.0.1}{12} }$ | ${\href{/padicField/3.12.0.1}{12} }$ | R | R | ${\href{/padicField/11.3.0.1}{3} }^{4}$ | ${\href{/padicField/13.4.0.1}{4} }^{3}$ | ${\href{/padicField/17.12.0.1}{12} }$ | ${\href{/padicField/19.6.0.1}{6} }^{2}$ | ${\href{/padicField/23.12.0.1}{12} }$ | ${\href{/padicField/29.2.0.1}{2} }^{6}$ | ${\href{/padicField/31.3.0.1}{3} }^{4}$ | ${\href{/padicField/37.12.0.1}{12} }$ | ${\href{/padicField/41.1.0.1}{1} }^{12}$ | ${\href{/padicField/43.4.0.1}{4} }^{3}$ | ${\href{/padicField/47.12.0.1}{12} }$ | ${\href{/padicField/53.12.0.1}{12} }$ | ${\href{/padicField/59.6.0.1}{6} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(5\) | 5.12.9.2 | $x^{12} + 12 x^{10} + 12 x^{9} + 69 x^{8} + 108 x^{7} + 42 x^{6} - 396 x^{5} + 840 x^{4} + 252 x^{3} + 1476 x^{2} + 684 x + 1601$ | $4$ | $3$ | $9$ | $C_{12}$ | $[\ ]_{4}^{3}$ |
\(7\) | 7.12.8.1 | $x^{12} + 15 x^{10} + 40 x^{9} + 84 x^{8} + 120 x^{7} + 53 x^{6} + 414 x^{5} - 1293 x^{4} - 1830 x^{3} + 10968 x^{2} - 13836 x + 12004$ | $3$ | $4$ | $8$ | $C_{12}$ | $[\ ]_{3}^{4}$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
* | 1.5.4t1.a.a | $1$ | $ 5 $ | \(\Q(\zeta_{5})\) | $C_4$ (as 4T1) | $0$ | $-1$ |
* | 1.5.2t1.a.a | $1$ | $ 5 $ | \(\Q(\sqrt{5}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
* | 1.5.4t1.a.b | $1$ | $ 5 $ | \(\Q(\zeta_{5})\) | $C_4$ (as 4T1) | $0$ | $-1$ |
* | 1.7.3t1.a.a | $1$ | $ 7 $ | \(\Q(\zeta_{7})^+\) | $C_3$ (as 3T1) | $0$ | $1$ |
* | 1.35.12t1.a.a | $1$ | $ 5 \cdot 7 $ | 12.0.11259376953125.1 | $C_{12}$ (as 12T1) | $0$ | $-1$ |
* | 1.35.6t1.b.a | $1$ | $ 5 \cdot 7 $ | 6.6.300125.1 | $C_6$ (as 6T1) | $0$ | $1$ |
* | 1.35.12t1.a.b | $1$ | $ 5 \cdot 7 $ | 12.0.11259376953125.1 | $C_{12}$ (as 12T1) | $0$ | $-1$ |
* | 1.7.3t1.a.b | $1$ | $ 7 $ | \(\Q(\zeta_{7})^+\) | $C_3$ (as 3T1) | $0$ | $1$ |
* | 1.35.12t1.a.c | $1$ | $ 5 \cdot 7 $ | 12.0.11259376953125.1 | $C_{12}$ (as 12T1) | $0$ | $-1$ |
* | 1.35.6t1.b.b | $1$ | $ 5 \cdot 7 $ | 6.6.300125.1 | $C_6$ (as 6T1) | $0$ | $1$ |
* | 1.35.12t1.a.d | $1$ | $ 5 \cdot 7 $ | 12.0.11259376953125.1 | $C_{12}$ (as 12T1) | $0$ | $-1$ |