# Properties

 Label 2.2.380.1 Degree $2$ Signature $[2, 0]$ Discriminant $2^{2}\cdot 5\cdot 19$ Root discriminant $19.49$ Ramified primes $2, 5, 19$ Class number $2$ Class group $[2]$ Galois group $C_2$ (as 2T1)

# Related objects

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-95, 0, 1]);

sage: x = polygen(QQ); K.<a> = NumberField(x^2 - 95)

gp: K = bnfinit(x^2 - 95, 1)

## Normalizeddefining polynomial

$$x^{2} - 95$$

magma: DefiningPolynomial(K);

sage: K.defining_polynomial()

gp: K.pol

## Invariants

 Degree: $2$ magma: Degree(K);  sage: K.degree()  gp: poldegree(K.pol) Signature: $[2, 0]$ magma: Signature(K);  sage: K.signature()  gp: K.sign Discriminant: $$380=2^{2}\cdot 5\cdot 19$$ magma: Discriminant(K);  sage: K.disc()  gp: K.disc Root discriminant: $19.49$ magma: Abs(Discriminant(K))^(1/Degree(K));  sage: (K.disc().abs())^(1./K.degree())  gp: abs(K.disc)^(1/poldegree(K.pol)) Ramified primes: $2, 5, 19$ magma: PrimeDivisors(Discriminant(K));  sage: K.disc().support()  gp: factor(abs(K.disc))[,1]~ This field is Galois and abelian over $\Q$. Conductor: $$380=2^{2}\cdot 5\cdot 19$$ Dirichlet character group: $\lbrace$$\chi_{380}(1,·), \chi_{380}(379,·)$$\rbrace$ This is not a CM field.

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

$1$, $a$

magma: IntegralBasis(K);

sage: K.integral_basis()

gp: K.zk

## Class group and class number

$C_{2}$, which has order $2$

magma: ClassGroup(K);

sage: K.class_group().invariants()

gp: K.clgp

## Unit group

magma: UK, f := UnitGroup(K);

sage: UK = K.unit_group()

 Rank: $1$ magma: UnitRank(K);  sage: UK.rank()  gp: K.fu Torsion generator: $$-1$$ (order $2$) magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);  sage: UK.torsion_generator()  gp: K.tu[2] Fundamental unit: $$4 a + 39$$ magma: [K!f(g): g in Generators(UK)];  sage: UK.fundamental_units()  gp: K.fu Regulator: $$4.3565444206$$ magma: Regulator(K);  sage: K.regulator()  gp: K.reg

## Galois group

$C_2$ (as 2T1):

magma: GaloisGroup(K);

sage: K.galois_group(type='pari')

gp: polgalois(K.pol)

 A cyclic group of order 2 The 2 conjugacy class representatives for $C_2$ Character table for $C_2$

## Intermediate fields

 The extension is primitive: there are no intermediate fields between this field and $\Q$.

## Frobenius cycle types

 $p$ Cycle type 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 R ${\href{/LocalNumberField/3.2.0.1}{2} }$ R ${\href{/LocalNumberField/7.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/11.2.0.1}{2} }$ ${\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/17.2.0.1}{2} }$ R ${\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }$ ${\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }$ ${\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

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];

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]])

## Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$2$2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2] 55.2.1.1x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
$19$19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$

## Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
* 1.2e2_5_19.2t1.1c1$1$ $2^{2} \cdot 5 \cdot 19$ $x^{2} - 95$ $C_2$ (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 *.