↑

Properties

Label 14.0.61223838305...0000.1
Degree $14$
Signature $[0, 7]$
Discriminant $-\,2^{12}\cdot 3^{12}\cdot 5^{12}\cdot 7^{11}\cdot 17^{12}$
Root discriminant $965.56$
Ramified primes $2, 3, 5, 7, 17$
Class number $7$ (GRH)
Class group $[7]$ (GRH)
Galois group $F_7$ (as 14T4)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![3746235728, 3855152, -14136080, 2140096, 10712632, -3858484, -425726, 120355, 1337, -721, 329, -119, 35, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 7*x^13 + 35*x^12 - 119*x^11 + 329*x^10 - 721*x^9 + 1337*x^8 + 120355*x^7 - 425726*x^6 - 3858484*x^5 + 10712632*x^4 + 2140096*x^3 - 14136080*x^2 + 3855152*x + 3746235728)
 
gp: K = bnfinit(x^14 - 7*x^13 + 35*x^12 - 119*x^11 + 329*x^10 - 721*x^9 + 1337*x^8 + 120355*x^7 - 425726*x^6 - 3858484*x^5 + 10712632*x^4 + 2140096*x^3 - 14136080*x^2 + 3855152*x + 3746235728, 1)
 

Normalized defining polynomial

\( x^{14} - 7 x^{13} + 35 x^{12} - 119 x^{11} + 329 x^{10} - 721 x^{9} + 1337 x^{8} + 120355 x^{7} - 425726 x^{6} - 3858484 x^{5} + 10712632 x^{4} + 2140096 x^{3} - 14136080 x^{2} + 3855152 x + 3746235728 \)

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:  \(-612238383051472862311374717543000000000000=-\,2^{12}\cdot 3^{12}\cdot 5^{12}\cdot 7^{11}\cdot 17^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $965.56$
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:  $2, 3, 5, 7, 17$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{14} a^{6} - \frac{1}{7} a^{5} - \frac{3}{14} a^{4} + \frac{3}{7} a^{3} + \frac{1}{7} a^{2} - \frac{2}{7} a - \frac{3}{7}$, $\frac{1}{14} a^{7} - \frac{1}{2} a^{3} + \frac{1}{7}$, $\frac{1}{420} a^{8} - \frac{1}{105} a^{7} + \frac{1}{30} a^{6} - \frac{1}{15} a^{5} + \frac{7}{60} a^{4} - \frac{2}{15} a^{3} + \frac{2}{15} a^{2} - \frac{38}{105} a + \frac{19}{105}$, $\frac{1}{420} a^{9} - \frac{1}{210} a^{7} - \frac{1}{210} a^{6} - \frac{1}{140} a^{5} + \frac{1}{21} a^{4} + \frac{6}{35} a^{3} - \frac{33}{70} a^{2} + \frac{2}{105} a + \frac{16}{105}$, $\frac{1}{2940} a^{10} - \frac{1}{2940} a^{9} - \frac{1}{1470} a^{8} - \frac{3}{98} a^{7} - \frac{13}{420} a^{6} - \frac{1}{420} a^{5} - \frac{1}{30} a^{4} + \frac{15}{98} a^{3} + \frac{223}{1470} a^{2} + \frac{314}{735} a + \frac{179}{735}$, $\frac{1}{2940} a^{11} - \frac{1}{980} a^{9} - \frac{1}{2940} a^{8} + \frac{17}{588} a^{7} - \frac{1}{35} a^{6} - \frac{19}{420} a^{5} - \frac{229}{2940} a^{4} - \frac{1}{2} a^{3} + \frac{223}{490} a^{2} - \frac{47}{147} a - \frac{99}{245}$, $\frac{1}{5880} a^{12} - \frac{1}{5880} a^{10} + \frac{1}{1470} a^{9} - \frac{1}{1960} a^{8} - \frac{19}{735} a^{7} - \frac{1}{168} a^{6} - \frac{71}{490} a^{5} + \frac{17}{420} a^{4} + \frac{4}{245} a^{3} - \frac{72}{245} a^{2} - \frac{23}{245} a - \frac{248}{735}$, $\frac{1}{308453407833076234644892779647880} a^{13} - \frac{12326655145680030711029857757}{308453407833076234644892779647880} a^{12} + \frac{39922561963686044500536865699}{308453407833076234644892779647880} a^{11} - \frac{8080193911362985758843480871}{102817802611025411548297593215960} a^{10} - \frac{104391035892629855784931964867}{308453407833076234644892779647880} a^{9} - \frac{137618507189009207131613937701}{308453407833076234644892779647880} a^{8} + \frac{249859629800147755502706844943}{44064772547582319234984682806840} a^{7} + \frac{1222268579383250437997182146229}{61690681566615246928978555929576} a^{6} - \frac{11334603551687359117261308505831}{51408901305512705774148796607980} a^{5} + \frac{2261857874126340606699004816225}{30845340783307623464489277964788} a^{4} - \frac{3681329334485235934190601360884}{38556675979134529330611597455985} a^{3} + \frac{17629628712681151121047936106743}{77113351958269058661223194911970} a^{2} - \frac{12594881579559526890938434774736}{38556675979134529330611597455985} a - \frac{18312141384506946649739016787}{786870938349684272053297907265}$

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

Class group and class number

$C_{7}$, which has order $7$ (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:  \( 795109679454562.6 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$F_7$ (as 14T4):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 42
The 7 conjugacy class representatives for $F_7$
Character table for $F_7$

Intermediate fields

\(\Q(\sqrt{-7}) \), 7.1.295740809071407000000.1

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

Sibling fields

Galois closure: data not computed
Degree 7 sibling: data not computed
Degree 21 sibling: 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 ${\href{/LocalNumberField/11.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{7}$ R ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/43.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }$

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$2.7.6.1$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$$[\ ]_{7}^{6}$
$5$5.14.12.1$x^{14} - 5 x^{7} + 50$$7$$2$$12$$F_7$$[\ ]_{7}^{6}$
$7$7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.6.5.5$x^{6} + 56$$6$$1$$5$$C_6$$[\ ]_{6}$
7.6.5.5$x^{6} + 56$$6$$1$$5$$C_6$$[\ ]_{6}$
$17$17.14.12.1$x^{14} - 17 x^{7} + 867$$7$$2$$12$$F_7$$[\ ]_{7}^{6}$