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Properties

Label 14.0.39818280550...0000.1
Degree $14$
Signature $[0, 7]$
Discriminant $-\,2^{12}\cdot 3^{12}\cdot 5^{7}\cdot 7^{14}\cdot 11^{13}$
Root discriminant $673.87$
Ramified primes $2, 3, 5, 7, 11$
Class number $588$ (GRH)
Class group $[7, 84]$ (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![107616980, -56224420, 61475764, -22376228, 15613444, -4496800, 1940554, -425645, 138677, -22561, 5621, -623, 119, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 7*x^13 + 119*x^12 - 623*x^11 + 5621*x^10 - 22561*x^9 + 138677*x^8 - 425645*x^7 + 1940554*x^6 - 4496800*x^5 + 15613444*x^4 - 22376228*x^3 + 61475764*x^2 - 56224420*x + 107616980)
 
gp: K = bnfinit(x^14 - 7*x^13 + 119*x^12 - 623*x^11 + 5621*x^10 - 22561*x^9 + 138677*x^8 - 425645*x^7 + 1940554*x^6 - 4496800*x^5 + 15613444*x^4 - 22376228*x^3 + 61475764*x^2 - 56224420*x + 107616980, 1)
 

Normalized defining polynomial

\( x^{14} - 7 x^{13} + 119 x^{12} - 623 x^{11} + 5621 x^{10} - 22561 x^{9} + 138677 x^{8} - 425645 x^{7} + 1940554 x^{6} - 4496800 x^{5} + 15613444 x^{4} - 22376228 x^{3} + 61475764 x^{2} - 56224420 x + 107616980 \)

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:  \(-3981828055054226410930351572089280000000=-\,2^{12}\cdot 3^{12}\cdot 5^{7}\cdot 7^{14}\cdot 11^{13}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $673.87$
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:  $2, 3, 5, 7, 11$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{11} a^{7} + \frac{2}{11} a^{6} - \frac{3}{11} a^{5} - \frac{3}{11} a^{4} - \frac{4}{11} a^{3} - \frac{1}{11} a^{2} + \frac{2}{11} a + \frac{3}{11}$, $\frac{1}{22} a^{8} + \frac{2}{11} a^{6} - \frac{4}{11} a^{5} - \frac{9}{22} a^{4} - \frac{2}{11} a^{3} + \frac{2}{11} a^{2} + \frac{5}{11} a - \frac{3}{11}$, $\frac{1}{22} a^{9} + \frac{3}{11} a^{6} + \frac{3}{22} a^{5} + \frac{4}{11} a^{4} - \frac{1}{11} a^{3} - \frac{4}{11} a^{2} + \frac{4}{11} a + \frac{5}{11}$, $\frac{1}{22} a^{10} - \frac{9}{22} a^{6} + \frac{2}{11} a^{5} - \frac{3}{11} a^{4} - \frac{3}{11} a^{3} - \frac{4}{11} a^{2} - \frac{1}{11} a + \frac{2}{11}$, $\frac{1}{22} a^{11} - \frac{1}{22} a^{7} - \frac{1}{11} a^{6} - \frac{4}{11} a^{5} - \frac{4}{11} a^{4} + \frac{2}{11} a^{3} - \frac{5}{11} a^{2} - \frac{1}{11} a + \frac{1}{11}$, $\frac{1}{242} a^{12} + \frac{5}{242} a^{11} - \frac{1}{22} a^{7} + \frac{4}{11} a^{6} - \frac{1}{11} a^{5} - \frac{5}{22} a^{4} - \frac{4}{11} a^{3} - \frac{1}{11} a^{2} - \frac{36}{121} a + \frac{40}{121}$, $\frac{1}{481965620955056589851030762411246} a^{13} + \frac{44894995202334474759225126262}{21907528225229844993228671018693} a^{12} - \frac{3646940901763828043669404493111}{481965620955056589851030762411246} a^{11} + \frac{189567708847846105272691712612}{21907528225229844993228671018693} a^{10} - \frac{239522238824241141790880188830}{21907528225229844993228671018693} a^{9} - \frac{44862171122785008135471936908}{21907528225229844993228671018693} a^{8} + \frac{1175065914923309684995214179847}{43815056450459689986457342037386} a^{7} - \frac{8714618498137899140551249317449}{21907528225229844993228671018693} a^{6} - \frac{8402532670762721181661236162261}{43815056450459689986457342037386} a^{5} - \frac{4068305304639442212700746871836}{21907528225229844993228671018693} a^{4} + \frac{4033633538765433796070311060997}{21907528225229844993228671018693} a^{3} + \frac{92490950542508811813594094527207}{240982810477528294925515381205623} a^{2} + \frac{671046407317335377963425571438}{21907528225229844993228671018693} a + \frac{50329928240629617580534767256857}{240982810477528294925515381205623}$

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

Class group and class number

$C_{7}\times C_{84}$, which has order $588$ (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:  \( -1 \) (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:  \( 854604676346.2672 \) (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

\(\Q(\sqrt{-55}) \), 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 R R R ${\href{/LocalNumberField/13.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }$ ${\href{/LocalNumberField/29.14.0.1}{14} }$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/43.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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$2.7.6.1$x^{7} - 2$$7$$1$$6$$C_7:C_3$$[\ ]_{7}^{3}$
2.7.6.1$x^{7} - 2$$7$$1$$6$$C_7:C_3$$[\ ]_{7}^{3}$
$3$3.14.12.1$x^{14} - 3 x^{7} + 18$$7$$2$$12$$F_7$$[\ ]_{7}^{6}$
$5$5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.12.6.1$x^{12} + 500 x^{6} - 3125 x^{2} + 62500$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
$7$7.7.7.5$x^{7} + 7 x + 7$$7$$1$$7$$F_7$$[7/6]_{6}$
7.7.7.5$x^{7} + 7 x + 7$$7$$1$$7$$F_7$$[7/6]_{6}$
$11$11.14.13.2$x^{14} + 33$$14$$1$$13$$(C_7:C_3) \times C_2$$[\ ]_{14}^{3}$