# 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)

# Related objects

Show commands: SageMath / Pari/GP / Magma

## Normalizeddefining 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: $$11259376953125$$ 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: $$5$$, $$7$$ 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: $$\frac{14}{181} a^{11} - \frac{42}{181} a^{10} + \frac{56}{181} a^{9} - \frac{126}{181} a^{8} + \frac{193}{181} a^{7} - \frac{168}{181} a^{6} - \frac{14}{181} a^{5} - \frac{350}{181} a^{4} + \frac{154}{181} a^{3} - \frac{980}{181} a^{2} + \frac{28}{181} a - \frac{14}{181}$$ (14)/(181)*a^(11) - (42)/(181)*a^(10) + (56)/(181)*a^(9) - (126)/(181)*a^(8) + (193)/(181)*a^(7) - (168)/(181)*a^(6) - (14)/(181)*a^(5) - (350)/(181)*a^(4) + (154)/(181)*a^(3) - (980)/(181)*a^(2) + (28)/(181)*a - (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: $\frac{9}{181}a^{11}+\frac{155}{181}a^{6}-\frac{404}{181}a+1$, $\frac{28}{181}a^{11}-\frac{84}{181}a^{10}+\frac{112}{181}a^{9}-\frac{252}{181}a^{8}+\frac{386}{181}a^{7}-\frac{336}{181}a^{6}-\frac{28}{181}a^{5}-\frac{700}{181}a^{4}+\frac{308}{181}a^{3}-\frac{1779}{181}a^{2}+\frac{56}{181}a-\frac{28}{181}$, $\frac{9}{181}a^{11}-\frac{27}{181}a^{10}+\frac{36}{181}a^{9}-\frac{81}{181}a^{8}+\frac{137}{181}a^{7}-\frac{108}{181}a^{6}-\frac{9}{181}a^{5}-\frac{225}{181}a^{4}+\frac{99}{181}a^{3}-\frac{449}{181}a^{2}+\frac{18}{181}a-\frac{9}{181}$, $\frac{88}{181}a^{11}-\frac{74}{181}a^{10}+\frac{241}{181}a^{9}-\frac{316}{181}a^{8}+\frac{711}{181}a^{7}+\frac{313}{181}a^{6}+\frac{1014}{181}a^{5}+\frac{168}{181}a^{4}+\frac{1975}{181}a^{3}-\frac{869}{181}a^{2}-\frac{9}{181}a-\frac{302}{181}$, $\frac{88}{181}a^{11}-\frac{70}{181}a^{10}+\frac{241}{181}a^{9}-\frac{316}{181}a^{8}+\frac{711}{181}a^{7}+\frac{313}{181}a^{6}+\frac{1103}{181}a^{5}+\frac{168}{181}a^{4}+\frac{1975}{181}a^{3}-\frac{869}{181}a^{2}-\frac{9}{181}a-\frac{381}{181}$ 9/181*a^11 + 155/181*a^6 - 404/181*a + 1, 28/181*a^11 - 84/181*a^10 + 112/181*a^9 - 252/181*a^8 + 386/181*a^7 - 336/181*a^6 - 28/181*a^5 - 700/181*a^4 + 308/181*a^3 - 1779/181*a^2 + 56/181*a - 28/181, 9/181*a^11 - 27/181*a^10 + 36/181*a^9 - 81/181*a^8 + 137/181*a^7 - 108/181*a^6 - 9/181*a^5 - 225/181*a^4 + 99/181*a^3 - 449/181*a^2 + 18/181*a - 9/181, 88/181*a^11 - 74/181*a^10 + 241/181*a^9 - 316/181*a^8 + 711/181*a^7 + 313/181*a^6 + 1014/181*a^5 + 168/181*a^4 + 1975/181*a^3 - 869/181*a^2 - 9/181*a - 302/181, 88/181*a^11 - 70/181*a^10 + 241/181*a^9 - 316/181*a^8 + 711/181*a^7 + 313/181*a^6 + 1103/181*a^5 + 168/181*a^4 + 1975/181*a^3 - 869/181*a^2 - 9/181*a - 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

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

## 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

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
$$5$$ 5.12.9.2$x^{12} - 10 x^{8} + 25 x^{4} - 500$$4$$3$$9$$C_{12}$$[\ ]_{4}^{3} $$7$$ 7.12.8.1x^{12} - 63 x^{9} + 637 x^{6} + 6174 x^{3} + 300125$$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 *.