# Properties

 Label 4.4.69696.2 Degree $4$ Signature $[4, 0]$ Discriminant $69696$ Root discriminant $16.25$ Ramified primes $2, 3, 11$ Class number $1$ Class group trivial Galois group $C_2^2$ (as 4T2)

# Related objects

Show commands for: SageMath / Pari/GP / Magma

## Normalizeddefining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^4 - 2*x^3 - 19*x^2 + 20*x + 34)

gp: K = bnfinit(x^4 - 2*x^3 - 19*x^2 + 20*x + 34, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![34, 20, -19, -2, 1]);

$$x^{4} - 2 x^{3} - 19 x^{2} + 20 x + 34$$

sage: K.defining_polynomial()

gp: K.pol

magma: DefiningPolynomial(K);

## Invariants

 Degree: $4$ sage: K.degree()  gp: poldegree(K.pol)  magma: Degree(K); Signature: $[4, 0]$ sage: K.signature()  gp: K.sign  magma: Signature(K); Discriminant: $$69696$$$$\medspace = 2^{6}\cdot 3^{2}\cdot 11^{2}$$ sage: K.disc()  gp: K.disc  magma: Discriminant(Integers(K)); Root discriminant: $16.25$ 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: $2, 3, 11$ sage: K.disc().support()  gp: factor(abs(K.disc))[,1]~  magma: PrimeDivisors(Discriminant(Integers(K))); $|\Gal(K/\Q)|$: $4$ This field is Galois and abelian over $\Q$. Conductor: $$264=2^{3}\cdot 3\cdot 11$$ Dirichlet character group: $\lbrace$$\chi_{264}(1,·), \chi_{264}(197,·), \chi_{264}(65,·), \chi_{264}(133,·)$$\rbrace$ This is not a CM field.

## Integral basis (with respect to field generator $$a$$)

$1$, $a$, $a^{2}$, $\frac{1}{25} a^{3} + \frac{11}{25} a^{2} - \frac{1}{25} a + \frac{7}{25}$

sage: K.integral_basis()

gp: K.zk

magma: IntegralBasis(K);

## 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: $3$ sage: UK.rank()  gp: K.fu  magma: UnitRank(K); Torsion generator: $$-1$$ (order $2$) sage: UK.torsion_generator()  gp: K.tu[2]  magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K); Fundamental units: $$\frac{2}{25} a^{3} - \frac{3}{25} a^{2} - \frac{27}{25} a + \frac{39}{25}$$,  $$\frac{4}{25} a^{3} - \frac{6}{25} a^{2} - \frac{4}{25} a + \frac{3}{25}$$,  $$\frac{16}{25} a^{3} - \frac{24}{25} a^{2} - \frac{416}{25} a - \frac{363}{25}$$ sage: UK.fundamental_units()  gp: K.fu  magma: [K!f(g): g in Generators(UK)]; Regulator: $$32.8461765739$$ sage: K.regulator()  gp: K.reg  magma: Regulator(K);

## Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{4}\cdot(2\pi)^{0}\cdot 32.8461765739 \cdot 1}{2\sqrt{69696}}\approx 0.995338684058$

## Galois group

$C_2^2$ (as 4T2):

sage: K.galois_group(type='pari')

gp: polgalois(K.pol)

magma: GaloisGroup(K);

 An abelian group of order 4 The 4 conjugacy class representatives for $C_2^2$ Character table for $C_2^2$

## Intermediate fields

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

## Multiplicative Galois module structure

 $U_{K^{gal}}/\textrm{Tors}(U_{K^{gal}}) \cong$ $A_1$ $\oplus$ $A_2$ $\oplus$ $A_3$ Galois action is Type I (ii)

## Frobenius cycle types

 $p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$ Cycle type R R ${\href{/LocalNumberField/5.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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
$2$2.2.3.1$x^{2} + 14$$2$$1$$3$$C_2$$[3] 2.2.3.1x^{2} + 14$$2$$1$$3$$C_2$$[3]$
$3$3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2} 1111.4.2.1x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$

## Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ $x$ $C_1$ $1$ $1$
* 1.33.2t1.a.a$1$ $3 \cdot 11$ $x^{2} - x - 8$ $C_2$ (as 2T1) $1$ $1$
* 1.264.2t1.a.a$1$ $2^{3} \cdot 3 \cdot 11$ $x^{2} - 66$ $C_2$ (as 2T1) $1$ $1$
* 1.8.2t1.a.a$1$ $2^{3}$ $x^{2} - 2$ $C_2$ (as 2T1) $1$ $1$

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 *.