magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 2, 0, 0, 2, 0, -1, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^7 - x^6 + 2*x^4 + 2*x + 1)
gp: K = bnfinit(x^7 - x^6 + 2*x^4 + 2*x + 1, 1)
Normalized defining polynomial
\( x^{7} - x^{6} + 2 x^{4} + 2 x + 1 \)
magma: DefiningPolynomial(K);
sage: K.defining_polynomial()
gp: K.pol
Invariants
| Degree: | $7$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[1, 3]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(-364871=-\,13^{2}\cdot 17\cdot 127\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $6.23$ | 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: | $13, 17, 127$ | 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: | $3$ | 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: | \( a^{6} - a^{5} + 2 a^{3} - a^{2} + a + 2 \), \( a \), \( a^{6} - 2 a^{5} + a^{4} + 2 a^{3} - 2 a^{2} + 2 \) | magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
| |
| Regulator: | \( 0.631311355925 \) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
$S_7$ (as 7T7):
magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
| A non-solvable group of order 5040 |
| The 15 conjugacy class representatives for $S_7$ |
| Character table for $S_7$ |
Intermediate fields
| The extension is primitive: there are no intermediate fields between this field and $\Q$. |
Sibling fields
| Degree 14 sibling: | Deg 14 |
| Degree 21 sibling: | Deg 21 |
| Degree 30 sibling: | data not computed |
| 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 | ${\href{/LocalNumberField/2.7.0.1}{7} }$ | ${\href{/LocalNumberField/3.7.0.1}{7} }$ | ${\href{/LocalNumberField/5.4.0.1}{4} }{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }$ | ${\href{/LocalNumberField/7.7.0.1}{7} }$ | ${\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }$ | R | R | ${\href{/LocalNumberField/19.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ | ${\href{/LocalNumberField/23.7.0.1}{7} }$ | ${\href{/LocalNumberField/29.7.0.1}{7} }$ | ${\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ | ${\href{/LocalNumberField/37.6.0.1}{6} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ | ${\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ | ${\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ | ${\href{/LocalNumberField/47.5.0.1}{5} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }$ | ${\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{3}$ |
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 |
|---|---|---|---|---|---|---|---|
| $13$ | $\Q_{13}$ | $x + 2$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| 13.2.0.1 | $x^{2} - x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 13.4.2.2 | $x^{4} - 13 x^{2} + 338$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ | |
| $17$ | 17.2.1.1 | $x^{2} - 17$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 17.5.0.1 | $x^{5} - x + 6$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | |
| $127$ | $\Q_{127}$ | $x + 9$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| 127.2.1.1 | $x^{2} - 127$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 127.2.0.1 | $x^{2} - x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
Artin representations
| Label | Dimension | Conductor | Defining polynomial of Artin field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| * | 1.1.1t1.1c1 | $1$ | $1$ | $x$ | $C_1$ | $1$ | $1$ |
| 1.17_127.2t1.1c1 | $1$ | $ 17 \cdot 127 $ | $x^{2} - x + 540$ | $C_2$ (as 2T1) | $1$ | $-1$ | |
| 6.13e2_17e5_127e5.14t46.1c1 | $6$ | $ 13^{2} \cdot 17^{5} \cdot 127^{5}$ | $x^{7} - x^{6} + 2 x^{4} + 2 x + 1$ | $S_7$ (as 7T7) | $1$ | $0$ | |
| * | 6.13e2_17_127.7t7.1c1 | $6$ | $ 13^{2} \cdot 17 \cdot 127 $ | $x^{7} - x^{6} + 2 x^{4} + 2 x + 1$ | $S_7$ (as 7T7) | $1$ | $0$ |
| 14.13e6_17e4_127e4.21t38.1c1 | $14$ | $ 13^{6} \cdot 17^{4} \cdot 127^{4}$ | $x^{7} - x^{6} + 2 x^{4} + 2 x + 1$ | $S_7$ (as 7T7) | $1$ | $2$ | |
| 14.13e6_17e10_127e10.42t413.1c1 | $14$ | $ 13^{6} \cdot 17^{10} \cdot 127^{10}$ | $x^{7} - x^{6} + 2 x^{4} + 2 x + 1$ | $S_7$ (as 7T7) | $1$ | $-2$ | |
| 14.13e6_17e9_127e9.30t565.1c1 | $14$ | $ 13^{6} \cdot 17^{9} \cdot 127^{9}$ | $x^{7} - x^{6} + 2 x^{4} + 2 x + 1$ | $S_7$ (as 7T7) | $1$ | $0$ | |
| 14.13e6_17e5_127e5.30t565.1c1 | $14$ | $ 13^{6} \cdot 17^{5} \cdot 127^{5}$ | $x^{7} - x^{6} + 2 x^{4} + 2 x + 1$ | $S_7$ (as 7T7) | $1$ | $0$ | |
| 15.13e8_17e5_127e5.42t412.1c1 | $15$ | $ 13^{8} \cdot 17^{5} \cdot 127^{5}$ | $x^{7} - x^{6} + 2 x^{4} + 2 x + 1$ | $S_7$ (as 7T7) | $1$ | $-3$ | |
| 15.13e8_17e10_127e10.42t411.1c1 | $15$ | $ 13^{8} \cdot 17^{10} \cdot 127^{10}$ | $x^{7} - x^{6} + 2 x^{4} + 2 x + 1$ | $S_7$ (as 7T7) | $1$ | $3$ | |
| 20.13e12_17e10_127e10.70.1c1 | $20$ | $ 13^{12} \cdot 17^{10} \cdot 127^{10}$ | $x^{7} - x^{6} + 2 x^{4} + 2 x + 1$ | $S_7$ (as 7T7) | $1$ | $0$ | |
| 21.13e10_17e10_127e10.84.1c1 | $21$ | $ 13^{10} \cdot 17^{10} \cdot 127^{10}$ | $x^{7} - x^{6} + 2 x^{4} + 2 x + 1$ | $S_7$ (as 7T7) | $1$ | $-3$ | |
| 21.13e10_17e11_127e11.42t418.1c1 | $21$ | $ 13^{10} \cdot 17^{11} \cdot 127^{11}$ | $x^{7} - x^{6} + 2 x^{4} + 2 x + 1$ | $S_7$ (as 7T7) | $1$ | $3$ | |
| 35.13e18_17e20_127e20.126.1c1 | $35$ | $ 13^{18} \cdot 17^{20} \cdot 127^{20}$ | $x^{7} - x^{6} + 2 x^{4} + 2 x + 1$ | $S_7$ (as 7T7) | $1$ | $-1$ | |
| 35.13e18_17e15_127e15.70.1c1 | $35$ | $ 13^{18} \cdot 17^{15} \cdot 127^{15}$ | $x^{7} - x^{6} + 2 x^{4} + 2 x + 1$ | $S_7$ (as 7T7) | $1$ | $1$ |
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 *.