Properties

Label 4.0.13026013.2
Degree $4$
Signature $[0, 2]$
Discriminant $7^{2}\cdot 11^{2}\cdot 13^{3}$
Root discriminant $60.08$
Ramified primes $7, 11, 13$
Class number $20$
Class group $[2, 10]$
Galois Group $C_4$ (as 4T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4943, 251, 249, -1, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^4 - x^3 + 249*x^2 + 251*x + 4943)
gp: K = bnfinit(x^4 - x^3 + 249*x^2 + 251*x + 4943, 1)

Normalized defining polynomial

\(x^{4} \) \(\mathstrut -\mathstrut x^{3} \) \(\mathstrut +\mathstrut 249 x^{2} \) \(\mathstrut +\mathstrut 251 x \) \(\mathstrut +\mathstrut 4943 \)

magma: DefiningPolynomial(K);
sage: K.defining_polynomial()
gp: K.pol

Invariants

Degree:  $4$
magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
Signature:  $[0, 2]$
magma: Signature(K);
sage: K.signature()
gp: K.sign
Discriminant:  \(13026013=7^{2}\cdot 11^{2}\cdot 13^{3}\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Root discriminant:  $60.08$
magma: Abs(Discriminant(K))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
Ramified primes:  $7, 11, 13$
magma: PrimeDivisors(Discriminant(K));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
This field is Galois and abelian over $\Q$.
Conductor:  \(1001=7\cdot 11\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{1001}(1,·)$, $\chi_{1001}(538,·)$, $\chi_{1001}(307,·)$, $\chi_{1001}(155,·)$$\rbrace$
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $\frac{1}{3461} a^{3} + \frac{57}{3461} a^{2} + \frac{94}{3461} a - \frac{1219}{3461}$

magma: IntegralBasis(K);
sage: K.integral_basis()
gp: K.zk

Class group and class number

$C_{2}\times C_{10}$, which has order $20$

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:  \( \frac{1}{3461} a^{3} + \frac{57}{3461} a^{2} + \frac{94}{3461} a + \frac{2242}{3461} \)
magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
Regulator:  \( 2.38952643457 \)
magma: Regulator(K);
sage: K.regulator()
gp: K.reg

Galois group

$C_4$ (as 4T1):

magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
A cyclic group of order 4
The 4 conjugacy class representatives for $C_4$
Character table for $C_4$

Intermediate fields

\(\Q(\sqrt{13}) \)

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 ${\href{/LocalNumberField/2.4.0.1}{4} }$ ${\href{/LocalNumberField/3.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/5.4.0.1}{4} }$ R R R ${\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }$ ${\href{/LocalNumberField/37.4.0.1}{4} }$ ${\href{/LocalNumberField/41.4.0.1}{4} }$ ${\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }$ ${\href{/LocalNumberField/53.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }$

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
$7$7.4.2.2$x^{4} - 7 x^{2} + 147$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
$11$11.4.2.2$x^{4} - 11 x^{2} + 847$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
$13$13.4.3.2$x^{4} - 52$$4$$1$$3$$C_4$$[\ ]_{4}$

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
* 1.7_11_13.4t1.1c1$1$ $ 7 \cdot 11 \cdot 13 $ $x^{4} - x^{3} + 249 x^{2} + 251 x + 4943$ $C_4$ (as 4T1) $0$ $-1$
* 1.13.2t1.1c1$1$ $ 13 $ $x^{2} - x - 3$ $C_2$ (as 2T1) $1$ $1$
* 1.7_11_13.4t1.1c2$1$ $ 7 \cdot 11 \cdot 13 $ $x^{4} - x^{3} + 249 x^{2} + 251 x + 4943$ $C_4$ (as 4T1) $0$ $-1$

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 *.