Properties

Label 12.0.11259376953125.1
Degree $12$
Signature $[0, 6]$
Discriminant $1.126\times 10^{13}$
Root discriminant \(12.24\)
Ramified primes see page
Class number $1$
Class group trivial
Galois group $C_{12}$ (as 12T1)

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Show commands: SageMath / Pari/GP / Magma

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]);
 

\( x^{12} - x^{11} + 3x^{10} - 4x^{9} + 9x^{8} + 2x^{7} + 12x^{6} + x^{5} + 25x^{4} - 11x^{3} + 5x^{2} - 2x + 1 \) Copy content Toggle raw display

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:   \(11259376953125\) \(\medspace = 5^{9}\cdot 7^{8}\) Copy content Toggle raw display
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:   \(5\), \(7\) Copy content Toggle raw display
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{181}a^{9}+\frac{43}{181}a^{8}+\frac{39}{181}a^{7}+\frac{48}{181}a^{6}+\frac{73}{181}a^{5}+\frac{39}{181}a^{4}+\frac{48}{181}a^{3}+\frac{73}{181}a^{2}+\frac{62}{181}a-\frac{49}{181}$, $\frac{1}{181}a^{10}-\frac{23}{181}a^{5}-\frac{65}{181}$, $\frac{1}{181}a^{11}-\frac{23}{181}a^{6}-\frac{65}{181}a$ Copy content Toggle raw display

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 $10$) Copy content Toggle raw display
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
Fundamental units:   $\frac{9}{181}a^{11}+\frac{155}{181}a^{6}-\frac{404}{181}a+1$, $\frac{28}{181}a^{11}-\frac{84}{181}a^{10}+\frac{112}{181}a^{9}-\frac{252}{181}a^{8}+\frac{386}{181}a^{7}-\frac{336}{181}a^{6}-\frac{28}{181}a^{5}-\frac{700}{181}a^{4}+\frac{308}{181}a^{3}-\frac{1779}{181}a^{2}+\frac{56}{181}a-\frac{28}{181}$, $\frac{9}{181}a^{11}-\frac{27}{181}a^{10}+\frac{36}{181}a^{9}-\frac{81}{181}a^{8}+\frac{137}{181}a^{7}-\frac{108}{181}a^{6}-\frac{9}{181}a^{5}-\frac{225}{181}a^{4}+\frac{99}{181}a^{3}-\frac{449}{181}a^{2}+\frac{18}{181}a-\frac{9}{181}$, $\frac{88}{181}a^{11}-\frac{74}{181}a^{10}+\frac{241}{181}a^{9}-\frac{316}{181}a^{8}+\frac{711}{181}a^{7}+\frac{313}{181}a^{6}+\frac{1014}{181}a^{5}+\frac{168}{181}a^{4}+\frac{1975}{181}a^{3}-\frac{869}{181}a^{2}-\frac{9}{181}a-\frac{302}{181}$, $\frac{88}{181}a^{11}-\frac{70}{181}a^{10}+\frac{241}{181}a^{9}-\frac{316}{181}a^{8}+\frac{711}{181}a^{7}+\frac{313}{181}a^{6}+\frac{1103}{181}a^{5}+\frac{168}{181}a^{4}+\frac{1975}{181}a^{3}-\frac{869}{181}a^{2}-\frac{9}{181}a-\frac{381}{181}$ Copy content Toggle raw display
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

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{0}\cdot(2\pi)^{6}\cdot 104.882003477 \cdot 1}{10\sqrt{11259376953125}}\approx 0.192319360106$

Galois group

$C_{12}$ (as 12T1):

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

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
\(5\) Copy content Toggle raw display 5.12.9.2$x^{12} - 10 x^{8} + 25 x^{4} - 500$$4$$3$$9$$C_{12}$$[\ ]_{4}^{3}$
\(7\) Copy content Toggle raw display 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 *.