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Properties

Label 14.14.1030159830...0000.1
Degree $14$
Signature $[14, 0]$
Discriminant $2^{16}\cdot 5^{12}\cdot 7^{14}\cdot 37^{7}$
Root discriminant $373.55$
Ramified primes $2, 5, 7, 37$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 14T37

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-207900, -3350900, -14154175, -11484900, 4518150, 6881000, 537775, -1102840, -227640, 49420, 13615, -420, -210, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 210*x^12 - 420*x^11 + 13615*x^10 + 49420*x^9 - 227640*x^8 - 1102840*x^7 + 537775*x^6 + 6881000*x^5 + 4518150*x^4 - 11484900*x^3 - 14154175*x^2 - 3350900*x - 207900)
 
gp: K = bnfinit(x^14 - 210*x^12 - 420*x^11 + 13615*x^10 + 49420*x^9 - 227640*x^8 - 1102840*x^7 + 537775*x^6 + 6881000*x^5 + 4518150*x^4 - 11484900*x^3 - 14154175*x^2 - 3350900*x - 207900, 1)
 

Normalized defining polynomial

\( x^{14} - 210 x^{12} - 420 x^{11} + 13615 x^{10} + 49420 x^{9} - 227640 x^{8} - 1102840 x^{7} + 537775 x^{6} + 6881000 x^{5} + 4518150 x^{4} - 11484900 x^{3} - 14154175 x^{2} - 3350900 x - 207900 \)

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

Invariants

Degree:  $14$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[14, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(1030159830727471620190672000000000000=2^{16}\cdot 5^{12}\cdot 7^{14}\cdot 37^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $373.55$
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:  $2, 5, 7, 37$
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 \(a\))

$1$, $a$, $a^{2}$, $\frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{4} a^{4} - \frac{1}{4} a^{3} + \frac{1}{4} a^{2} + \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{4} a^{5} - \frac{1}{2} a^{2} - \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{8} a^{6} - \frac{1}{8} a^{5} - \frac{1}{4} a^{3} + \frac{1}{8} a^{2} + \frac{3}{8} a - \frac{1}{4}$, $\frac{1}{80} a^{7} - \frac{1}{16} a^{5} - \frac{1}{8} a^{4} + \frac{3}{16} a^{3} - \frac{1}{4} a^{2} + \frac{1}{16} a + \frac{3}{8}$, $\frac{1}{80} a^{8} - \frac{1}{16} a^{6} - \frac{1}{8} a^{5} - \frac{1}{16} a^{4} - \frac{3}{16} a^{2} + \frac{1}{8} a - \frac{1}{2}$, $\frac{1}{160} a^{9} - \frac{1}{16} a^{6} - \frac{1}{16} a^{5} + \frac{1}{16} a^{4} - \frac{7}{16} a^{2} - \frac{11}{32} a + \frac{7}{16}$, $\frac{1}{160} a^{10} - \frac{1}{16} a^{6} - \frac{1}{8} a^{4} + \frac{13}{32} a^{2} + \frac{3}{8}$, $\frac{1}{320} a^{11} - \frac{1}{320} a^{10} - \frac{1}{160} a^{7} + \frac{1}{32} a^{6} + \frac{1}{16} a^{5} + \frac{1}{16} a^{4} - \frac{11}{64} a^{3} - \frac{29}{64} a^{2} + \frac{5}{16} a - \frac{7}{16}$, $\frac{1}{320} a^{12} - \frac{1}{320} a^{10} - \frac{1}{160} a^{8} - \frac{1}{32} a^{6} - \frac{1}{8} a^{5} - \frac{7}{64} a^{4} - \frac{1}{64} a^{2} + \frac{1}{8} a - \frac{7}{16}$, $\frac{1}{6035641554950368385229440} a^{13} + \frac{7636609982751266081313}{6035641554950368385229440} a^{12} - \frac{5572039278633471246657}{6035641554950368385229440} a^{11} + \frac{18755795477728002058407}{6035641554950368385229440} a^{10} - \frac{1226099914905326019999}{3017820777475184192614720} a^{9} + \frac{459340551696340566943}{603564155495036838522944} a^{8} - \frac{9649018410686220856741}{3017820777475184192614720} a^{7} - \frac{23384866090119172834149}{603564155495036838522944} a^{6} - \frac{134738774113902146101799}{1207128310990073677045888} a^{5} - \frac{43319184922449136355255}{1207128310990073677045888} a^{4} + \frac{118037214952843553301279}{1207128310990073677045888} a^{3} + \frac{218648392234354398069279}{1207128310990073677045888} a^{2} - \frac{29988193171208122065405}{75445519436879604815368} a - \frac{41801856212899635421953}{301782077747518419261472}$

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:  $13$
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:  \( 94986945469500 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

14T37:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1764
The 25 conjugacy class representatives for [1/2.F_42(7)^2]2
Character table for [1/2.F_42(7)^2]2 is not computed

Intermediate fields

\(\Q(\sqrt{37}) \)

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 ${\href{/LocalNumberField/3.7.0.1}{7} }{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }$ R R ${\href{/LocalNumberField/11.6.0.1}{6} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/13.14.0.1}{14} }$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }$ ${\href{/LocalNumberField/29.14.0.1}{14} }$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }$ R ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.14.0.1}{14} }$ ${\href{/LocalNumberField/47.7.0.1}{7} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.7.0.1}{7} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }$

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$2.2.0.1$x^{2} - x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.12.16.3$x^{12} - 30 x^{10} - 5 x^{8} + 19 x^{4} + 30 x^{2} + 1$$6$$2$$16$$C_6\times S_3$$[2]_{3}^{6}$
$5$5.14.12.1$x^{14} - 5 x^{7} + 50$$7$$2$$12$$F_7$$[\ ]_{7}^{6}$
$7$7.7.7.3$x^{7} + 35 x + 7$$7$$1$$7$$F_7$$[7/6]_{6}$
7.7.7.2$x^{7} + 21 x + 7$$7$$1$$7$$F_7$$[7/6]_{6}$
37Data not computed