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Properties

Label 14.2.50612760739...0000.1
Degree $14$
Signature $[2, 6]$
Discriminant $2^{27}\cdot 5^{7}\cdot 13^{6}$
Root discriminant $25.55$
Ramified primes $2, 5, 13$
Class number $2$
Class group $[2]$
Galois group $D_7^2$ (as 14T13)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![64, -96, 68, 24, -72, 28, 89, -176, 106, 8, -50, 24, 2, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 4*x^13 + 2*x^12 + 24*x^11 - 50*x^10 + 8*x^9 + 106*x^8 - 176*x^7 + 89*x^6 + 28*x^5 - 72*x^4 + 24*x^3 + 68*x^2 - 96*x + 64)
 
gp: K = bnfinit(x^14 - 4*x^13 + 2*x^12 + 24*x^11 - 50*x^10 + 8*x^9 + 106*x^8 - 176*x^7 + 89*x^6 + 28*x^5 - 72*x^4 + 24*x^3 + 68*x^2 - 96*x + 64, 1)
 

Normalized defining polynomial

\( x^{14} - 4 x^{13} + 2 x^{12} + 24 x^{11} - 50 x^{10} + 8 x^{9} + 106 x^{8} - 176 x^{7} + 89 x^{6} + 28 x^{5} - 72 x^{4} + 24 x^{3} + 68 x^{2} - 96 x + 64 \)

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:  \(50612760739840000000=2^{27}\cdot 5^{7}\cdot 13^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $25.55$
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:  $2, 5, 13$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4}$, $\frac{1}{4} a^{9} - \frac{1}{4} a^{7} - \frac{1}{4} a^{5} - \frac{1}{4} a^{3} - \frac{1}{2} a$, $\frac{1}{8} a^{10} - \frac{1}{8} a^{9} - \frac{1}{8} a^{8} - \frac{1}{8} a^{7} + \frac{1}{8} a^{6} - \frac{3}{8} a^{5} + \frac{3}{8} a^{4} + \frac{3}{8} a^{3} - \frac{1}{2} a^{2} + \frac{1}{4} a$, $\frac{1}{8} a^{11} - \frac{1}{4} a^{8} - \frac{1}{4} a^{7} - \frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} - \frac{3}{8} a^{3} - \frac{1}{4} a^{2} - \frac{1}{4} a$, $\frac{1}{80} a^{12} + \frac{1}{20} a^{11} - \frac{1}{40} a^{10} - \frac{3}{40} a^{9} + \frac{1}{20} a^{8} + \frac{3}{40} a^{7} + \frac{9}{40} a^{5} + \frac{3}{80} a^{4} - \frac{3}{40} a^{3} - \frac{19}{40} a^{2} + \frac{1}{5} a - \frac{1}{5}$, $\frac{1}{155662400} a^{13} - \frac{32211}{7783120} a^{12} + \frac{4599221}{77831200} a^{11} + \frac{483029}{19457800} a^{10} - \frac{2087101}{77831200} a^{9} - \frac{46783}{972890} a^{8} - \frac{19421567}{77831200} a^{7} + \frac{411061}{19457800} a^{6} - \frac{67525919}{155662400} a^{5} - \frac{11303}{108400} a^{4} + \frac{6553177}{19457800} a^{3} - \frac{412793}{2432225} a^{2} - \frac{3347303}{7783120} a + \frac{3868149}{9728900}$

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

Class group and class number

$C_{2}$, which has order $2$

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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 114268.201269 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_7^2$ (as 14T13):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 196
The 25 conjugacy class representatives for $D_7^2$
Character table for $D_7^2$ is not computed

Intermediate fields

\(\Q(\sqrt{10}) \)

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

Sibling fields

Degree 14 siblings: data not computed
Degree 28 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 ${\href{/LocalNumberField/3.7.0.1}{7} }^{2}$ R ${\href{/LocalNumberField/7.14.0.1}{14} }$ ${\href{/LocalNumberField/11.14.0.1}{14} }$ R ${\href{/LocalNumberField/17.14.0.1}{14} }$ ${\href{/LocalNumberField/19.14.0.1}{14} }$ ${\href{/LocalNumberField/23.14.0.1}{14} }$ ${\href{/LocalNumberField/29.14.0.1}{14} }$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/43.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/47.14.0.1}{14} }$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.14.0.1}{14} }$

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$2.2.3.2$x^{2} + 6$$2$$1$$3$$C_2$$[3]$
2.4.8.4$x^{4} + 6 x^{2} + 4 x + 6$$4$$1$$8$$C_2^2$$[2, 3]$
2.4.8.4$x^{4} + 6 x^{2} + 4 x + 6$$4$$1$$8$$C_2^2$$[2, 3]$
2.4.8.4$x^{4} + 6 x^{2} + 4 x + 6$$4$$1$$8$$C_2^2$$[2, 3]$
$5$5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$13$$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.7.6.1$x^{7} - 13$$7$$1$$6$$D_{7}$$[\ ]_{7}^{2}$