Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - x^11 + 3*x^10 - 4*x^9 + 9*x^8 + 2*x^7 + 12*x^6 + x^5 + 25*x^4 - 11*x^3 + 5*x^2 - 2*x + 1)
gp: K = bnfinit(x^12 - x^11 + 3*x^10 - 4*x^9 + 9*x^8 + 2*x^7 + 12*x^6 + x^5 + 25*x^4 - 11*x^3 + 5*x^2 - 2*x + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -2, 5, -11, 25, 1, 12, 2, 9, -4, 3, -1, 1]);
\( x^{12} - x^{11} + 3x^{10} - 4x^{9} + 9x^{8} + 2x^{7} + 12x^{6} + x^{5} + 25x^{4} - 11x^{3} + 5x^{2} - 2x + 1 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
Invariants
| Degree: | $12$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
| |
| Signature: | $[0, 6]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
| |
| Discriminant: |
\(11259376953125\)
\(\medspace = 5^{9}\cdot 7^{8}\)
| sage: K.disc()
gp: K.disc
magma: Discriminant(Integers(K));
| |
| Root discriminant: | \(12.24\) | sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
| |
| Ramified primes: |
\(5\), \(7\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(Integers(K)));
| |
| $\card{ \Gal(K/\Q) }$: | $12$ | ||
| 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. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{181}a^{9}+\frac{43}{181}a^{8}+\frac{39}{181}a^{7}+\frac{48}{181}a^{6}+\frac{73}{181}a^{5}+\frac{39}{181}a^{4}+\frac{48}{181}a^{3}+\frac{73}{181}a^{2}+\frac{62}{181}a-\frac{49}{181}$, $\frac{1}{181}a^{10}-\frac{23}{181}a^{5}-\frac{65}{181}$, $\frac{1}{181}a^{11}-\frac{23}{181}a^{6}-\frac{65}{181}a$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$
sage: K.class_group().invariants()
gp: K.clgp
magma: ClassGroup(K);
Unit group
sage: UK = K.unit_group()
magma: UK, f := UnitGroup(K);
| Rank: | $5$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
| |
| 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);
| |
| 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!f(g): g in Generators(UK)];
| |
| Regulator: | \( 104.882003477 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
|
Class number formula
Galois group
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
magma: GaloisGroup(K);
| 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.
sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
gp: idealfactors = idealprimedec(K, p); \\ get the data
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])
magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
magma: idealfactors := Factorization(p*Integers(K)); // get the data
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
Local algebras for ramified primes
| $p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
|---|---|---|---|---|---|---|---|
|
\(5\)
| 5.12.9.2 | $x^{12} - 10 x^{8} + 25 x^{4} - 500$ | $4$ | $3$ | $9$ | $C_{12}$ | $[\ ]_{4}^{3}$ |
|
\(7\)
| 7.12.8.1 | $x^{12} - 63 x^{9} + 637 x^{6} + 6174 x^{3} + 300125$ | $3$ | $4$ | $8$ | $C_{12}$ | $[\ ]_{3}^{4}$ |
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) | $0$ | $-1$ |
Data is given for all irreducible
representations of the Galois group for the Galois closure
of this field. Those marked with * are summands in the
permutation representation coming from this field. Representations
which appear with multiplicity greater than one are indicated
by exponents on the *.