# Properties

 Label 7.7.128100283921.1 Degree $7$ Signature $[7, 0]$ Discriminant $128100283921$ Root discriminant $38.62$ Ramified prime $71$ Class number $1$ Class group trivial Galois group $C_7$ (as 7T1)

# Related objects

Show commands for: SageMath / Pari/GP / Magma

## Normalizeddefining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^7 - x^6 - 30*x^5 - 3*x^4 + 254*x^3 + 246*x^2 - 245*x - 137)

gp: K = bnfinit(x^7 - x^6 - 30*x^5 - 3*x^4 + 254*x^3 + 246*x^2 - 245*x - 137, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-137, -245, 246, 254, -3, -30, -1, 1]);

$$x^{7} - x^{6} - 30 x^{5} - 3 x^{4} + 254 x^{3} + 246 x^{2} - 245 x - 137$$

sage: K.defining_polynomial()

gp: K.pol

magma: DefiningPolynomial(K);

## Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ $x$ $C_1$ $1$ $1$
* 1.71.7t1.a.a$1$ $71$ $x^{7} - x^{6} - 30 x^{5} - 3 x^{4} + 254 x^{3} + 246 x^{2} - 245 x - 137$ $C_7$ (as 7T1) $0$ $1$
* 1.71.7t1.a.b$1$ $71$ $x^{7} - x^{6} - 30 x^{5} - 3 x^{4} + 254 x^{3} + 246 x^{2} - 245 x - 137$ $C_7$ (as 7T1) $0$ $1$
* 1.71.7t1.a.c$1$ $71$ $x^{7} - x^{6} - 30 x^{5} - 3 x^{4} + 254 x^{3} + 246 x^{2} - 245 x - 137$ $C_7$ (as 7T1) $0$ $1$
* 1.71.7t1.a.d$1$ $71$ $x^{7} - x^{6} - 30 x^{5} - 3 x^{4} + 254 x^{3} + 246 x^{2} - 245 x - 137$ $C_7$ (as 7T1) $0$ $1$
* 1.71.7t1.a.e$1$ $71$ $x^{7} - x^{6} - 30 x^{5} - 3 x^{4} + 254 x^{3} + 246 x^{2} - 245 x - 137$ $C_7$ (as 7T1) $0$ $1$
* 1.71.7t1.a.f$1$ $71$ $x^{7} - x^{6} - 30 x^{5} - 3 x^{4} + 254 x^{3} + 246 x^{2} - 245 x - 137$ $C_7$ (as 7T1) $0$ $1$

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 *.