magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![19, -122, 345, -591, 709, -647, 472, -274, 138, -71, 40, -22, 12, -5, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 5*x^13 + 12*x^12 - 22*x^11 + 40*x^10 - 71*x^9 + 138*x^8 - 274*x^7 + 472*x^6 - 647*x^5 + 709*x^4 - 591*x^3 + 345*x^2 - 122*x + 19)
gp: K = bnfinit(x^14 - 5*x^13 + 12*x^12 - 22*x^11 + 40*x^10 - 71*x^9 + 138*x^8 - 274*x^7 + 472*x^6 - 647*x^5 + 709*x^4 - 591*x^3 + 345*x^2 - 122*x + 19, 1)
Normalized defining polynomial
\( x^{14} - 5 x^{13} + 12 x^{12} - 22 x^{11} + 40 x^{10} - 71 x^{9} + 138 x^{8} - 274 x^{7} + 472 x^{6} - 647 x^{5} + 709 x^{4} - 591 x^{3} + 345 x^{2} - 122 x + 19 \)
magma: DefiningPolynomial(K);
sage: K.defining_polynomial()
gp: K.pol
Invariants
| Degree: | $14$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[0, 7]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(-347455511790057216=-\,2^{8}\cdot 3^{7}\cdot 7^{10}\cdot 13^{3}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $17.90$ | 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: | $2, 3, 7, 13$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{938749622} a^{13} + \frac{61118873}{469374811} a^{12} - \frac{124389450}{469374811} a^{11} + \frac{129467968}{469374811} a^{10} - \frac{187016943}{469374811} a^{9} - \frac{417622593}{938749622} a^{8} + \frac{8999105}{55220566} a^{7} + \frac{255629253}{938749622} a^{6} + \frac{251269263}{938749622} a^{5} + \frac{131717046}{469374811} a^{4} - \frac{329543901}{938749622} a^{3} + \frac{184063221}{469374811} a^{2} + \frac{243136239}{938749622} a - \frac{164466991}{938749622}$
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: | $6$ | magma: UnitRank(K);
sage: UK.rank()
gp: K.fu
| |
| Torsion generator: | \( -\frac{8526761}{9114074} a^{13} + \frac{18840361}{4557037} a^{12} - \frac{40367530}{4557037} a^{11} + \frac{70937609}{4557037} a^{10} - \frac{130440559}{4557037} a^{9} + \frac{457436521}{9114074} a^{8} - \frac{53979085}{536122} a^{7} + \frac{1815182197}{9114074} a^{6} - \frac{2994738779}{9114074} a^{5} + \frac{1912869077}{4557037} a^{4} - \frac{3899720269}{9114074} a^{3} + \frac{1429041167}{4557037} a^{2} - \frac{1353431015}{9114074} a + \frac{298212719}{9114074} \) (order $6$) | magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
sage: UK.torsion_generator()
gp: K.tu[2]
| |
| Fundamental units: | Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right | magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
| |
| Regulator: | \( 5507.6213891 \) | 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 solvable group of order 3528 |
| The 35 conjugacy class representatives for [F_42(7)^2]2=F_42(7)wr2 |
| Character table for [F_42(7)^2]2=F_42(7)wr2 is not computed |
Intermediate fields
| \(\Q(\sqrt{-3}) \) |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
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 | ${\href{/LocalNumberField/5.12.0.1}{12} }{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }$ | R | ${\href{/LocalNumberField/11.12.0.1}{12} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }$ | R | ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }$ | ${\href{/LocalNumberField/19.7.0.1}{7} }{,}\,{\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ | ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }$ | ${\href{/LocalNumberField/29.14.0.1}{14} }$ | ${\href{/LocalNumberField/31.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/37.7.0.1}{7} }{,}\,{\href{/LocalNumberField/37.6.0.1}{6} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ | ${\href{/LocalNumberField/41.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }$ | ${\href{/LocalNumberField/43.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }$ | ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }$ | ${\href{/LocalNumberField/59.12.0.1}{12} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }$ |
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
| $p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
|---|---|---|---|---|---|---|---|
| $2$ | 2.2.0.1 | $x^{2} - x + 1$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 2.12.8.2 | $x^{12} - 8 x^{3} + 16$ | $3$ | $4$ | $8$ | $C_3\times (C_3 : C_4)$ | $[\ ]_{3}^{12}$ | |
| $3$ | 3.2.1.2 | $x^{2} + 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 3.6.3.2 | $x^{6} - 9 x^{2} + 27$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
| 3.6.3.2 | $x^{6} - 9 x^{2} + 27$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
| $7$ | $\Q_{7}$ | $x + 2$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| 7.6.3.2 | $x^{6} - 49 x^{2} + 686$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
| 7.7.7.3 | $x^{7} + 35 x + 7$ | $7$ | $1$ | $7$ | $F_7$ | $[7/6]_{6}$ | |
| $13$ | $\Q_{13}$ | $x + 2$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{13}$ | $x + 2$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 13.3.0.1 | $x^{3} - 2 x + 6$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| 13.3.0.1 | $x^{3} - 2 x + 6$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| 13.6.3.1 | $x^{6} - 52 x^{4} + 676 x^{2} - 79092$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ |