magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2, 2, 0, 0, 0, 0, -2, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^7 - 2*x^6 + 2*x + 2)
gp: K = bnfinit(x^7 - 2*x^6 + 2*x + 2, 1)
Normalized defining polynomial
\( x^{7} - 2 x^{6} + 2 x + 2 \)
magma: DefiningPolynomial(K);
sage: K.defining_polynomial()
gp: K.pol
Invariants
| Degree: | $7$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[3, 2]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(50808384=2^{6}\cdot 3^{8}\cdot 11^{2}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $12.61$ | magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
| |
| Ramified primes: | $2, 3, 11$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is not Galois over $\Q$. | |||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$
magma: IntegralBasis(K);
sage: K.integral_basis()
gp: K.zk
Class group and class number
Trivial group, which has order $1$
magma: ClassGroup(K);
sage: K.class_group().invariants()
gp: K.clgp
Unit group
magma: UK, f := UnitGroup(K);
sage: UK = K.unit_group()
| Rank: | $4$ | 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 units: | \( a^{6} - 2 a^{5} + a^{4} - a^{3} - a + 1 \), \( a^{3} - a - 1 \), \( a^{3} - a^{2} - a - 1 \), \( a^{6} - 3 a^{5} + 3 a^{4} - 3 a^{3} + 2 a^{2} - a + 3 \) | magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
| |
| Regulator: | \( 38.0037816148 \) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
$A_7$ (as 7T6):
magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
| A non-solvable group of order 2520 |
| The 9 conjugacy class representatives for $A_7$ |
| Character table for $A_7$ |
Intermediate fields
| The extension is primitive: there are no intermediate fields between this field and $\Q$. |
Sibling fields
| Degree 15 siblings: | Deg 15, Deg 15 |
| Degree 21 sibling: | Deg 21 |
| Degree 35 sibling: | data not computed |
| Degree 42 siblings: | data not computed |
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 | ${\href{/LocalNumberField/5.7.0.1}{7} }$ | ${\href{/LocalNumberField/7.7.0.1}{7} }$ | R | ${\href{/LocalNumberField/13.3.0.1}{3} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ | ${\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ | ${\href{/LocalNumberField/23.7.0.1}{7} }$ | ${\href{/LocalNumberField/29.7.0.1}{7} }$ | ${\href{/LocalNumberField/31.7.0.1}{7} }$ | ${\href{/LocalNumberField/37.7.0.1}{7} }$ | ${\href{/LocalNumberField/41.5.0.1}{5} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/43.5.0.1}{5} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/47.7.0.1}{7} }$ | ${\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ | ${\href{/LocalNumberField/59.7.0.1}{7} }$ |
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
| $p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
|---|---|---|---|---|---|---|---|
| $2$ | 2.7.6.1 | $x^{7} - 2$ | $7$ | $1$ | $6$ | $C_7:C_3$ | $[\ ]_{7}^{3}$ |
| $3$ | 3.3.5.3 | $x^{3} + 12$ | $3$ | $1$ | $5$ | $S_3$ | $[5/2]_{2}$ |
| 3.4.3.2 | $x^{4} - 3$ | $4$ | $1$ | $3$ | $D_{4}$ | $[\ ]_{4}^{2}$ | |
| $11$ | 11.3.2.1 | $x^{3} - 11$ | $3$ | $1$ | $2$ | $S_3$ | $[\ ]_{3}^{2}$ |
| 11.4.0.1 | $x^{4} - x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |
Artin representations
| Label | Dimension | Conductor | Defining polynomial of Artin field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| * | 1.1.1t1.1c1 | $1$ | $1$ | $x$ | $C_1$ | $1$ | $1$ |
| * | 6.2e6_3e8_11e2.7t6.1c1 | $6$ | $ 2^{6} \cdot 3^{8} \cdot 11^{2}$ | $x^{7} - 2 x^{6} + 2 x + 2$ | $A_7$ (as 7T6) | $1$ | $2$ |
| 10.2e9_3e18_11e6.70.1c1 | $10$ | $ 2^{9} \cdot 3^{18} \cdot 11^{6}$ | $x^{7} - 2 x^{6} + 2 x + 2$ | $A_7$ (as 7T6) | $0$ | $-2$ | |
| 10.2e9_3e18_11e6.70.1c2 | $10$ | $ 2^{9} \cdot 3^{18} \cdot 11^{6}$ | $x^{7} - 2 x^{6} + 2 x + 2$ | $A_7$ (as 7T6) | $0$ | $-2$ | |
| 14.2e12_3e28_11e10.15t47.1c1 | $14$ | $ 2^{12} \cdot 3^{28} \cdot 11^{10}$ | $x^{7} - 2 x^{6} + 2 x + 2$ | $A_7$ (as 7T6) | $1$ | $2$ | |
| 14.2e12_3e24_11e8.21t33.1c1 | $14$ | $ 2^{12} \cdot 3^{24} \cdot 11^{8}$ | $x^{7} - 2 x^{6} + 2 x + 2$ | $A_7$ (as 7T6) | $1$ | $2$ | |
| 15.2e12_3e26_11e8.42t294.1c1 | $15$ | $ 2^{12} \cdot 3^{26} \cdot 11^{8}$ | $x^{7} - 2 x^{6} + 2 x + 2$ | $A_7$ (as 7T6) | $1$ | $-1$ | |
| 21.2e18_3e44_11e16.42t299.1c1 | $21$ | $ 2^{18} \cdot 3^{44} \cdot 11^{16}$ | $x^{7} - 2 x^{6} + 2 x + 2$ | $A_7$ (as 7T6) | $1$ | $1$ | |
| 35.2e30_3e68_11e24.70.1c1 | $35$ | $ 2^{30} \cdot 3^{68} \cdot 11^{24}$ | $x^{7} - 2 x^{6} + 2 x + 2$ | $A_7$ (as 7T6) | $1$ | $-1$ |
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 *.