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Properties

Label 14.2.22328753169...7264.1
Degree $14$
Signature $[2, 6]$
Discriminant $2^{20}\cdot 3^{13}\cdot 7^{14}\cdot 11^{12}\cdot 13^{7}$
Root discriminant $1471.55$
Ramified primes $2, 3, 7, 11, 13$
Class number $168$ (GRH)
Class group $[2, 2, 42]$ (GRH)
Galois group $F_7 \times C_2$ (as 14T7)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-137230989255, -109621512, 24631206327, -14054040, -1894708179, -216216, 80970435, -264, -2076165, 0, 31941, 0, -273, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 273*x^12 + 31941*x^10 - 2076165*x^8 - 264*x^7 + 80970435*x^6 - 216216*x^5 - 1894708179*x^4 - 14054040*x^3 + 24631206327*x^2 - 109621512*x - 137230989255)
 
gp: K = bnfinit(x^14 - 273*x^12 + 31941*x^10 - 2076165*x^8 - 264*x^7 + 80970435*x^6 - 216216*x^5 - 1894708179*x^4 - 14054040*x^3 + 24631206327*x^2 - 109621512*x - 137230989255, 1)
 

Normalized defining polynomial

\( x^{14} - 273 x^{12} + 31941 x^{10} - 2076165 x^{8} - 264 x^{7} + 80970435 x^{6} - 216216 x^{5} - 1894708179 x^{4} - 14054040 x^{3} + 24631206327 x^{2} - 109621512 x - 137230989255 \)

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

Invariants

Degree:  $14$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[2, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(223287531694546552453480881226223313579147264=2^{20}\cdot 3^{13}\cdot 7^{14}\cdot 11^{12}\cdot 13^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $1471.55$
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:  $2, 3, 7, 11, 13$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{2}$, $\frac{1}{12} a^{7} - \frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} + \frac{1}{4} a^{3} + \frac{1}{4} a^{2} + \frac{1}{4} a + \frac{1}{4}$, $\frac{1}{12} a^{8} + \frac{1}{4}$, $\frac{1}{12} a^{9} + \frac{1}{4} a$, $\frac{1}{24} a^{10} - \frac{1}{24} a^{8} - \frac{1}{4} a^{6} - \frac{1}{4} a^{4} - \frac{1}{2} a^{3} + \frac{3}{8} a^{2} - \frac{1}{2} a + \frac{1}{8}$, $\frac{1}{24} a^{11} - \frac{1}{24} a^{9} - \frac{1}{4} a^{6} - \frac{1}{4} a^{4} + \frac{1}{8} a^{3} - \frac{1}{4} a^{2} - \frac{1}{8} a - \frac{1}{4}$, $\frac{1}{72} a^{12} - \frac{1}{24} a^{8} + \frac{1}{6} a^{5} - \frac{1}{8} a^{4} - \frac{1}{2} a + \frac{3}{8}$, $\frac{1}{274070079639583478273834023815976338718776} a^{13} + \frac{1167140368157162035668032198447572895107}{274070079639583478273834023815976338718776} a^{12} - \frac{717550299700327938887637890481154202281}{91356693213194492757944674605325446239592} a^{11} + \frac{246235736039533884040509957528682009565}{91356693213194492757944674605325446239592} a^{10} - \frac{586574107364534746451641465391299660087}{15226115535532415459657445767554241039932} a^{9} + \frac{1400728326095102764264473890907184418347}{45678346606597246378972337302662723119796} a^{8} - \frac{144522115716728289858993689806741609531}{11419586651649311594743084325665680779949} a^{7} + \frac{3649550094971542899942416851364542929631}{22839173303298623189486168651331361559898} a^{6} + \frac{9279854964893820438253153116976553301805}{91356693213194492757944674605325446239592} a^{5} + \frac{6033964072715199424287688698465367159113}{30452231071064830919314891535108482079864} a^{4} + \frac{14818212582535201690424012514490634253403}{30452231071064830919314891535108482079864} a^{3} - \frac{10817288700549745278155264652337183267363}{30452231071064830919314891535108482079864} a^{2} - \frac{1611082167989028299242617460976974125537}{15226115535532415459657445767554241039932} a - \frac{4641037702319632393528824760483180640839}{15226115535532415459657445767554241039932}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{42}$, which has order $168$ (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:  $7$
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:  \( 918103443441014.5 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times F_7$ (as 14T7):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 84
The 14 conjugacy class representatives for $F_7 \times C_2$
Character table for $F_7 \times C_2$

Intermediate fields

\(\Q(\sqrt{39}) \), 7.1.68069081958026688.23

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

Sibling fields

Degree 14 sibling: data not computed
Degree 28 sibling: 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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{2}$ R R R ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.14.0.1}{14} }$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.14.0.1}{14} }$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }$

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

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$2$2.14.20.8$x^{14} + 2 x^{13} + 2 x^{11} + 2 x^{8} + 2 x^{7} + 2 x^{6} + 2$$14$$1$$20$$(C_7:C_3) \times C_2$$[2]_{7}^{3}$
$3$3.14.13.1$x^{14} - 3$$14$$1$$13$$F_7 \times C_2$$[\ ]_{14}^{6}$
$7$7.7.7.5$x^{7} + 7 x + 7$$7$$1$$7$$F_7$$[7/6]_{6}$
7.7.7.5$x^{7} + 7 x + 7$$7$$1$$7$$F_7$$[7/6]_{6}$
$11$11.14.12.1$x^{14} - 11 x^{7} + 847$$7$$2$$12$$(C_7:C_3) \times C_2$$[\ ]_{7}^{6}$
$13$13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$