# Properties

 Label 2.2.140.1 Degree 2 Signature $[2, 0]$ Discriminant $2^{2}\cdot 5\cdot 7$ Ramified primes $2, 5, 7$ Class number 2 Class group [2] Galois Group $C_2$

# Related objects

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-35, 0, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^2 - 35)
gp: K = bnfinit(x^2 - 35, 1)

## Normalizeddefining polynomial

$x^{2}$ $\mathstrut -\mathstrut 35$

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: $140=2^{2}\cdot 5\cdot 7$ magma: Discriminant(K); sage: K.disc() gp: K.disc Ramified primes: $2, 5, 7$ magma: PrimeDivisors(Discriminant(K)); sage: K.disc().support() gp: factor(abs(K.disc))[,1]~ This field is Galois and abelian over $\Q$. Conductor: $140=2^{2}\cdot 5\cdot 7$ Dirichlet group: $\lbrace$$\chi_{140}(1,·), \chi_{140}(139,·)$$\rbrace$

## 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

Multiplicative Abelian group isomorphic to C2, 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: $a + 6$ magma: [K!f(g): g in Generators(UK)]; sage: UK.fundamental_units() gp: K.fu Regulator: $2.47788873029$ magma: Regulator(K); sage: K.regulator() gp: K.reg

## Galois group

magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
 A cyclic group of order 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 R ${\href{/LocalNumberField/11.2.0.1}{2} }$ ${\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }$ ${\href{/LocalNumberField/41.2.0.1}{2} }$ ${\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }$ ${\href{/LocalNumberField/53.2.0.1}{2} }$ ${\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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$2.2.2.2$x^{2} + 2 x - 2$$2$$1$$2$$C_2$$[2] 55.2.1.2x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
$7$7.2.1.2$x^{2} + 14$$2$$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_7.2t1.1c1$1$ $2^{2} \cdot 5 \cdot 7$ $x^{2} - 35$ $C_2$ $1$ $1$

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 *.