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Properties

Label 14.14.3852371860...0000.1
Degree $14$
Signature $[14, 0]$
Discriminant $2^{8}\cdot 3^{4}\cdot 5^{7}\cdot 61^{2}\cdot 389^{2}\cdot 649871^{2}$
Root discriminant $129.80$
Ramified primes $2, 3, 5, 61, 389, 649871$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 14T52

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-59399, 366349, -472530, -483750, 1020415, -327968, -197183, 123182, 563, -11496, 1303, 410, -68, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 5*x^13 - 68*x^12 + 410*x^11 + 1303*x^10 - 11496*x^9 + 563*x^8 + 123182*x^7 - 197183*x^6 - 327968*x^5 + 1020415*x^4 - 483750*x^3 - 472530*x^2 + 366349*x - 59399)
 
gp: K = bnfinit(x^14 - 5*x^13 - 68*x^12 + 410*x^11 + 1303*x^10 - 11496*x^9 + 563*x^8 + 123182*x^7 - 197183*x^6 - 327968*x^5 + 1020415*x^4 - 483750*x^3 - 472530*x^2 + 366349*x - 59399, 1)
 

Normalized defining polynomial

\( x^{14} - 5 x^{13} - 68 x^{12} + 410 x^{11} + 1303 x^{10} - 11496 x^{9} + 563 x^{8} + 123182 x^{7} - 197183 x^{6} - 327968 x^{5} + 1020415 x^{4} - 483750 x^{3} - 472530 x^{2} + 366349 x - 59399 \)

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:  \(385237186031186411963220000000=2^{8}\cdot 3^{4}\cdot 5^{7}\cdot 61^{2}\cdot 389^{2}\cdot 649871^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $129.80$
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:  $2, 3, 5, 61, 389, 649871$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{6} a^{10} - \frac{1}{6} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{6} a^{6} - \frac{1}{6} a^{4} - \frac{1}{2} a^{3} - \frac{1}{6} a^{2} - \frac{1}{6} a + \frac{1}{6}$, $\frac{1}{18} a^{11} + \frac{1}{18} a^{10} - \frac{1}{9} a^{9} - \frac{1}{6} a^{8} + \frac{1}{9} a^{7} + \frac{2}{9} a^{6} + \frac{1}{9} a^{5} + \frac{2}{9} a^{4} - \frac{1}{18} a^{3} + \frac{1}{3} a^{2} - \frac{1}{18} a + \frac{5}{18}$, $\frac{1}{18} a^{12} - \frac{2}{9} a^{9} - \frac{2}{9} a^{8} - \frac{7}{18} a^{7} - \frac{5}{18} a^{6} + \frac{1}{9} a^{5} - \frac{4}{9} a^{4} - \frac{1}{9} a^{3} + \frac{4}{9} a^{2} + \frac{1}{6} a - \frac{1}{9}$, $\frac{1}{8332313431671619058268} a^{13} + \frac{50932250241972307619}{2083078357917904764567} a^{12} - \frac{16144787453580514939}{694359452639301588189} a^{11} + \frac{28295470342267332152}{2083078357917904764567} a^{10} + \frac{40790681966325345325}{2777437810557206352756} a^{9} + \frac{1347171162435839499445}{2777437810557206352756} a^{8} - \frac{631444651384909023161}{4166156715835809529134} a^{7} - \frac{1029913595317710136094}{2083078357917904764567} a^{6} + \frac{2603667828638811412439}{8332313431671619058268} a^{5} - \frac{633583580598097843561}{2777437810557206352756} a^{4} - \frac{996391435383278574269}{2083078357917904764567} a^{3} + \frac{1705672198090146823583}{4166156715835809529134} a^{2} + \frac{43249646066390148775}{2083078357917904764567} a - \frac{2584284093359576618095}{8332313431671619058268}$

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:  \( 68466846022.4 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

14T52:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 56448
The 27 conjugacy class representatives for [L(7)^2]2=L(7)wr2
Character table for [L(7)^2]2=L(7)wr2 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \)

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

Sibling fields

Degree 16 sibling: data not computed
Degree 28 siblings: data not computed
Degree 42 sibling: data not computed
Arithmetically equvalently 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 R ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }$ ${\href{/LocalNumberField/11.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/13.14.0.1}{14} }$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.14.0.1}{14} }$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }$ ${\href{/LocalNumberField/41.7.0.1}{7} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ ${\href{/LocalNumberField/43.14.0.1}{14} }$ ${\href{/LocalNumberField/47.8.0.1}{8} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }$ ${\href{/LocalNumberField/59.7.0.1}{7} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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$2.2.0.1$x^{2} - x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.4.4.4$x^{4} - 5$$2$$2$$4$$D_{4}$$[2, 2]^{2}$
2.4.4.3$x^{4} + 2 x^{2} + 4 x + 4$$2$$2$$4$$D_{4}$$[2, 2]^{2}$
2.4.0.1$x^{4} - x + 1$$1$$4$$0$$C_4$$[\ ]^{4}$
$3$3.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
$5$5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$61$$\Q_{61}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{61}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{61}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{61}$$x + 2$$1$$1$$0$Trivial$[\ ]$
61.3.0.1$x^{3} - x + 10$$1$$3$$0$$C_3$$[\ ]^{3}$
61.3.0.1$x^{3} - x + 10$$1$$3$$0$$C_3$$[\ ]^{3}$
61.4.2.1$x^{4} + 183 x^{2} + 14884$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
389Data not computed
649871Data not computed