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Properties

Label 14.2.36202216328...1088.1
Degree $14$
Signature $[2, 6]$
Discriminant $2^{27}\cdot 7^{12}\cdot 11^{7}$
Root discriminant $66.93$
Ramified primes $2, 7, 11$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 14T25

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-6272, -4480, 18704, -21952, 22512, -23520, 15876, -5648, 840, -224, 196, -56, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 56*x^11 + 196*x^10 - 224*x^9 + 840*x^8 - 5648*x^7 + 15876*x^6 - 23520*x^5 + 22512*x^4 - 21952*x^3 + 18704*x^2 - 4480*x - 6272)
 
gp: K = bnfinit(x^14 - 56*x^11 + 196*x^10 - 224*x^9 + 840*x^8 - 5648*x^7 + 15876*x^6 - 23520*x^5 + 22512*x^4 - 21952*x^3 + 18704*x^2 - 4480*x - 6272, 1)
 

Normalized defining polynomial

\( x^{14} - 56 x^{11} + 196 x^{10} - 224 x^{9} + 840 x^{8} - 5648 x^{7} + 15876 x^{6} - 23520 x^{5} + 22512 x^{4} - 21952 x^{3} + 18704 x^{2} - 4480 x - 6272 \)

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:  \(36202216328934500847321088=2^{27}\cdot 7^{12}\cdot 11^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $66.93$
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:  $2, 7, 11$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4}$, $\frac{1}{2} a^{5}$, $\frac{1}{4} a^{6} - \frac{1}{2} a^{2}$, $\frac{1}{24} a^{7} - \frac{1}{6} a^{4} + \frac{1}{12} a^{3} + \frac{1}{3} a^{2} + \frac{1}{6} a - \frac{1}{3}$, $\frac{1}{48} a^{8} - \frac{1}{12} a^{5} + \frac{1}{24} a^{4} + \frac{1}{6} a^{3} - \frac{5}{12} a^{2} + \frac{1}{3} a$, $\frac{1}{96} a^{9} - \frac{1}{24} a^{6} + \frac{1}{48} a^{5} + \frac{1}{12} a^{4} - \frac{5}{24} a^{3} - \frac{1}{3} a^{2} - \frac{1}{2} a$, $\frac{1}{96} a^{10} + \frac{1}{48} a^{6} + \frac{1}{12} a^{5} + \frac{1}{8} a^{4} - \frac{1}{4} a^{3} - \frac{1}{6} a^{2} + \frac{1}{6} a - \frac{1}{3}$, $\frac{1}{96} a^{11} - \frac{1}{48} a^{7} + \frac{1}{12} a^{6} + \frac{1}{8} a^{5} - \frac{1}{12} a^{4} - \frac{1}{4} a^{3} - \frac{1}{6} a^{2} - \frac{1}{2} a + \frac{1}{3}$, $\frac{1}{192} a^{12} - \frac{1}{16} a^{6} + \frac{1}{6} a^{5} + \frac{1}{16} a^{4} + \frac{5}{12} a^{3} + \frac{11}{24} a^{2} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{38361535296} a^{13} - \frac{9597773}{5480219328} a^{12} - \frac{408291}{456684944} a^{11} - \frac{121511}{171256854} a^{10} + \frac{3324325}{2740109664} a^{9} + \frac{1808257}{456684944} a^{8} + \frac{1052777}{1370054832} a^{7} + \frac{937482331}{9590383824} a^{6} - \frac{72623155}{685027416} a^{5} + \frac{232845463}{1370054832} a^{4} - \frac{14240883}{114171236} a^{3} + \frac{77988275}{685027416} a^{2} - \frac{52038151}{171256854} a - \frac{3120328}{85628427}$

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:  $7$
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:  \( 650123158.139 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

14T25:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 588
The 19 conjugacy class representatives for [7^2:6_3]2
Character table for [7^2:6_3]2

Intermediate fields

\(\Q(\sqrt{22}) \)

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

Sibling fields

Degree 21 siblings: 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 ${\href{/LocalNumberField/3.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/5.14.0.1}{14} }$ R R ${\href{/LocalNumberField/13.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/17.14.0.1}{14} }$ ${\href{/LocalNumberField/19.14.0.1}{14} }$ ${\href{/LocalNumberField/23.14.0.1}{14} }$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.14.0.1}{14} }$ ${\href{/LocalNumberField/37.14.0.1}{14} }$ ${\href{/LocalNumberField/41.14.0.1}{14} }$ ${\href{/LocalNumberField/43.14.0.1}{14} }$ ${\href{/LocalNumberField/47.14.0.1}{14} }$ ${\href{/LocalNumberField/53.14.0.1}{14} }$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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$2.2.3.4$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]$
2.4.8.3$x^{4} + 6 x^{2} + 4 x + 14$$4$$1$$8$$C_2^2$$[2, 3]$
2.4.8.3$x^{4} + 6 x^{2} + 4 x + 14$$4$$1$$8$$C_2^2$$[2, 3]$
$7$$\Q_{7}$$x + 2$$1$$1$$0$Trivial$[\ ]$
7.6.5.6$x^{6} + 224$$6$$1$$5$$C_6$$[\ ]_{6}$
7.7.7.6$x^{7} + 28 x + 7$$7$$1$$7$$F_7$$[7/6]_{6}$
$11$11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$