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Properties

Label 14.14.7617290481...0400.1
Degree $14$
Signature $[14, 0]$
Discriminant $2^{31}\cdot 3^{11}\cdot 5^{2}\cdot 29^{2}\cdot 53^{2}\cdot 18413^{2}$
Root discriminant $160.64$
Ramified primes $2, 3, 5, 29, 53, 18413$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 14T58

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![538, -1260, -4049, 11724, -2158, -12656, 5850, 5232, -2712, -1076, 492, 112, -36, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 4*x^13 - 36*x^12 + 112*x^11 + 492*x^10 - 1076*x^9 - 2712*x^8 + 5232*x^7 + 5850*x^6 - 12656*x^5 - 2158*x^4 + 11724*x^3 - 4049*x^2 - 1260*x + 538)
 
gp: K = bnfinit(x^14 - 4*x^13 - 36*x^12 + 112*x^11 + 492*x^10 - 1076*x^9 - 2712*x^8 + 5232*x^7 + 5850*x^6 - 12656*x^5 - 2158*x^4 + 11724*x^3 - 4049*x^2 - 1260*x + 538, 1)
 

Normalized defining polynomial

\( x^{14} - 4 x^{13} - 36 x^{12} + 112 x^{11} + 492 x^{10} - 1076 x^{9} - 2712 x^{8} + 5232 x^{7} + 5850 x^{6} - 12656 x^{5} - 2158 x^{4} + 11724 x^{3} - 4049 x^{2} - 1260 x + 538 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $14$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[14, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(7617290481169169453542696550400=2^{31}\cdot 3^{11}\cdot 5^{2}\cdot 29^{2}\cdot 53^{2}\cdot 18413^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $160.64$
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:  $2, 3, 5, 29, 53, 18413$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{11607} a^{12} + \frac{1831}{3869} a^{11} - \frac{377}{11607} a^{10} - \frac{5023}{11607} a^{9} - \frac{1506}{3869} a^{8} + \frac{665}{3869} a^{7} - \frac{552}{3869} a^{6} + \frac{1516}{3869} a^{5} - \frac{181}{3869} a^{4} + \frac{26}{219} a^{3} + \frac{1163}{3869} a^{2} - \frac{1391}{11607} a + \frac{932}{11607}$, $\frac{1}{32604063} a^{13} + \frac{863}{32604063} a^{12} + \frac{7298773}{32604063} a^{11} - \frac{9657587}{32604063} a^{10} + \frac{14453866}{32604063} a^{9} + \frac{1812199}{10868021} a^{8} - \frac{1709876}{10868021} a^{7} + \frac{4522728}{10868021} a^{6} + \frac{977962}{10868021} a^{5} - \frac{7489766}{32604063} a^{4} - \frac{8547160}{32604063} a^{3} - \frac{13868882}{32604063} a^{2} + \frac{6127873}{32604063} a + \frac{15672094}{32604063}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

Trivial group, which has order $1$ (assuming GRH)

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $13$
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 (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 732906735238 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

14T58:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 12700800
The 54 conjugacy class representatives for [A(7)^2]2=A(7)wr2 are not computed
Character table for [A(7)^2]2=A(7)wr2 is not computed

Intermediate fields

\(\Q(\sqrt{6}) \)

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 30 siblings: data not computed
Degree 42 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 R R ${\href{/LocalNumberField/7.14.0.1}{14} }$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }$ ${\href{/LocalNumberField/13.14.0.1}{14} }$ ${\href{/LocalNumberField/17.8.0.1}{8} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }$ ${\href{/LocalNumberField/19.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/23.7.0.1}{7} }{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/31.14.0.1}{14} }$ ${\href{/LocalNumberField/37.8.0.1}{8} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }$ ${\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.7.0.1}{7} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.5.0.1}{5} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{3}$ R ${\href{/LocalNumberField/59.10.0.1}{10} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$2$2.2.3.4$x^{2} + 10$$2$$1$$3$$C_2$$[3]$
2.4.8.3$x^{4} + 6 x^{2} + 4 x + 14$$4$$1$$8$$C_2^2$$[2, 3]$
2.8.20.8$x^{8} + 4 x^{7} + 10 x^{4} + 4$$4$$2$$20$$D_4\times C_2$$[2, 3, 7/2]^{2}$
$3$3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.10.9.2$x^{10} + 3$$10$$1$$9$$F_{5}\times C_2$$[\ ]_{10}^{4}$
$5$$\Q_{5}$$x + 2$$1$$1$$0$Trivial$[\ ]$
5.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
5.4.2.2$x^{4} - 5 x^{2} + 50$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
5.7.0.1$x^{7} - x + 2$$1$$7$$0$$C_7$$[\ ]^{7}$
$29$$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.1$x^{2} - 29$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.7.0.1$x^{7} - x + 3$$1$$7$$0$$C_7$$[\ ]^{7}$
53Data not computed
18413Data not computed