magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 3, 12, 26, 35, 83, 9, -116, -38, -2, -23, 1, 1, -1, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - x^13 + x^12 + x^11 - 23*x^10 - 2*x^9 - 38*x^8 - 116*x^7 + 9*x^6 + 83*x^5 + 35*x^4 + 26*x^3 + 12*x^2 + 3*x + 1)
gp: K = bnfinit(x^14 - x^13 + x^12 + x^11 - 23*x^10 - 2*x^9 - 38*x^8 - 116*x^7 + 9*x^6 + 83*x^5 + 35*x^4 + 26*x^3 + 12*x^2 + 3*x + 1, 1)
Normalized defining polynomial
\( x^{14} - x^{13} + x^{12} + x^{11} - 23 x^{10} - 2 x^{9} - 38 x^{8} - 116 x^{7} + 9 x^{6} + 83 x^{5} + 35 x^{4} + 26 x^{3} + 12 x^{2} + 3 x + 1 \)
magma: DefiningPolynomial(K);
sage: K.defining_polynomial()
gp: K.pol
Invariants
| Degree: | $14$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[6, 4]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(7625019359376272152=2^{3}\cdot 13\cdot 557\cdot 11717\cdot 47869\cdot 234683\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $22.32$ | 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: | $2, 13, 557, 11717, 47869, 234683$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{31327660834} a^{13} + \frac{5509647382}{15663830417} a^{12} - \frac{13013167433}{31327660834} a^{11} + \frac{7238117775}{15663830417} a^{10} + \frac{11281218163}{31327660834} a^{9} - \frac{685714401}{31327660834} a^{8} - \frac{13558709673}{31327660834} a^{7} + \frac{10569631551}{31327660834} a^{6} - \frac{4679082511}{15663830417} a^{5} - \frac{13220299357}{31327660834} a^{4} + \frac{3609588784}{15663830417} a^{3} + \frac{4545368329}{15663830417} a^{2} + \frac{7513920791}{15663830417} a - \frac{9835706815}{31327660834}$
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: | $9$ | 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: | 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: | \( 50698.4507704 \) | 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 non-solvable group of order 87178291200 |
| The 135 conjugacy class representatives for S14 are not computed |
| Character table for S14 is not computed |
Intermediate fields
| The extension is primitive: there are no intermediate fields between this field and $\Q$. |
Sibling fields
| Degree 28 sibling: | 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 | ${\href{/LocalNumberField/3.8.0.1}{8} }{,}\,{\href{/LocalNumberField/3.6.0.1}{6} }$ | ${\href{/LocalNumberField/5.10.0.1}{10} }{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/7.9.0.1}{9} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ | ${\href{/LocalNumberField/11.6.0.1}{6} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }^{2}$ | R | ${\href{/LocalNumberField/17.6.0.1}{6} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ | ${\href{/LocalNumberField/19.11.0.1}{11} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ | ${\href{/LocalNumberField/23.9.0.1}{9} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ | ${\href{/LocalNumberField/29.7.0.1}{7} }{,}\,{\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ | ${\href{/LocalNumberField/31.7.0.1}{7} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ | ${\href{/LocalNumberField/37.12.0.1}{12} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }$ | ${\href{/LocalNumberField/41.11.0.1}{11} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ | ${\href{/LocalNumberField/43.5.0.1}{5} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }$ | ${\href{/LocalNumberField/47.7.0.1}{7} }{,}\,{\href{/LocalNumberField/47.6.0.1}{6} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ | ${\href{/LocalNumberField/53.11.0.1}{11} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }$ | ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }$ |
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 |
|---|---|---|---|---|---|---|---|
| $2$ | $\Q_{2}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| 2.2.3.3 | $x^{2} + 2$ | $2$ | $1$ | $3$ | $C_2$ | $[3]$ | |
| 2.4.0.1 | $x^{4} - x + 1$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 2.7.0.1 | $x^{7} - x + 1$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | |
| $13$ | 13.2.1.1 | $x^{2} - 13$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 13.3.0.1 | $x^{3} - 2 x + 6$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| 13.3.0.1 | $x^{3} - 2 x + 6$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| 13.6.0.1 | $x^{6} + x^{2} - 2 x + 2$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
| 557 | Data not computed | ||||||
| 11717 | Data not computed | ||||||
| 47869 | Data not computed | ||||||
| 234683 | Data not computed | ||||||