↑

Properties

Label 14.0.28772790931...5696.1
Degree $14$
Signature $[0, 7]$
Discriminant $-\,2^{8}\cdot 3^{7}\cdot 7^{12}\cdot 13^{5}$
Root discriminant $34.10$
Ramified primes $2, 3, 7, 13$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 14T45

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2092, 716, -393, -892, 1902, 99, -655, 16, 391, -136, -51, 29, 3, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 5*x^13 + 3*x^12 + 29*x^11 - 51*x^10 - 136*x^9 + 391*x^8 + 16*x^7 - 655*x^6 + 99*x^5 + 1902*x^4 - 892*x^3 - 393*x^2 + 716*x + 2092)
 
gp: K = bnfinit(x^14 - 5*x^13 + 3*x^12 + 29*x^11 - 51*x^10 - 136*x^9 + 391*x^8 + 16*x^7 - 655*x^6 + 99*x^5 + 1902*x^4 - 892*x^3 - 393*x^2 + 716*x + 2092, 1)
 

Normalized defining polynomial

\( x^{14} - 5 x^{13} + 3 x^{12} + 29 x^{11} - 51 x^{10} - 136 x^{9} + 391 x^{8} + 16 x^{7} - 655 x^{6} + 99 x^{5} + 1902 x^{4} - 892 x^{3} - 393 x^{2} + 716 x + 2092 \)

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:  \(-2877279093133463805696=-\,2^{8}\cdot 3^{7}\cdot 7^{12}\cdot 13^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $34.10$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{845760511666097882264} a^{13} - \frac{174865928226336680547}{845760511666097882264} a^{12} - \frac{132312717376138403827}{845760511666097882264} a^{11} - \frac{147306398167354367313}{845760511666097882264} a^{10} - \frac{190699417337371489933}{845760511666097882264} a^{9} + \frac{138544382325197216067}{422880255833048941132} a^{8} + \frac{127671944278046380699}{845760511666097882264} a^{7} - \frac{174872809805998986189}{422880255833048941132} a^{6} + \frac{188184632990834541573}{845760511666097882264} a^{5} - \frac{207956074056806331979}{845760511666097882264} a^{4} - \frac{18240301216979835677}{105720063958262235283} a^{3} - \frac{62674376934736361743}{211440127916524470566} a^{2} - \frac{280886663413162621969}{845760511666097882264} a + \frac{17477284520093069597}{422880255833048941132}$

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:  $6$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( \frac{52541144687}{301678826410328} a^{13} - \frac{381051961933}{301678826410328} a^{12} + \frac{727021822427}{301678826410328} a^{11} + \frac{1395515995633}{301678826410328} a^{10} - \frac{6819054703203}{301678826410328} a^{9} - \frac{152097730819}{150839413205164} a^{8} + \frac{39049796944061}{301678826410328} a^{7} - \frac{25951536600111}{150839413205164} a^{6} - \frac{43198608807405}{301678826410328} a^{5} + \frac{125370906491219}{301678826410328} a^{4} + \frac{2470592564614}{37709853301291} a^{3} - \frac{70870275633517}{75419706602582} a^{2} + \frac{40117728041009}{301678826410328} a + \frac{138836692177235}{150839413205164} \) (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 (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 663625.509498 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

14T45:

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

Degree 28 siblings: 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 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 ${\href{/LocalNumberField/17.14.0.1}{14} }$ ${\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.14.0.1}{14} }$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.7.0.1}{7} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.12.0.1}{12} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }$ ${\href{/LocalNumberField/43.7.0.1}{7} }{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }$

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

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$2$2.2.0.1$x^{2} - x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.12.8.1$x^{12} - 6 x^{9} + 12 x^{6} - 8 x^{3} + 16$$3$$4$$8$$C_3 : C_4$$[\ ]_{3}^{4}$
$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$Trivial$[\ ]$
7.6.5.2$x^{6} - 7$$6$$1$$5$$C_6$$[\ ]_{6}$
7.7.7.6$x^{7} + 28 x + 7$$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$$[\ ]^{3}$
13.6.5.1$x^{6} - 52$$6$$1$$5$$C_6$$[\ ]_{6}$