↑

Properties

Label 14.0.32918715676...4443.4
Degree $14$
Signature $[0, 7]$
Discriminant $-\,7^{24}\cdot 43^{13}$
Root discriminant $923.70$
Ramified primes $7, 43$
Class number $40187$ (GRH)
Class group $[40187]$ (GRH)
Galois group $C_{14}$ (as 14T1)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![20950836200249, 7534219954478, 375732658658, -206213160958, -15205020719, 4782561910, 285753447, -54303754, -4232963, 494242, 70735, -7826, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 7826*x^11 + 70735*x^10 + 494242*x^9 - 4232963*x^8 - 54303754*x^7 + 285753447*x^6 + 4782561910*x^5 - 15205020719*x^4 - 206213160958*x^3 + 375732658658*x^2 + 7534219954478*x + 20950836200249)
 
gp: K = bnfinit(x^14 - 7826*x^11 + 70735*x^10 + 494242*x^9 - 4232963*x^8 - 54303754*x^7 + 285753447*x^6 + 4782561910*x^5 - 15205020719*x^4 - 206213160958*x^3 + 375732658658*x^2 + 7534219954478*x + 20950836200249, 1)
 

Normalized defining polynomial

\( x^{14} - 7826 x^{11} + 70735 x^{10} + 494242 x^{9} - 4232963 x^{8} - 54303754 x^{7} + 285753447 x^{6} + 4782561910 x^{5} - 15205020719 x^{4} - 206213160958 x^{3} + 375732658658 x^{2} + 7534219954478 x + 20950836200249 \)

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:  \(-329187156767051876908796215815371233484443=-\,7^{24}\cdot 43^{13}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $923.70$
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:  $7, 43$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(2107=7^{2}\cdot 43\)
Dirichlet character group:    $\lbrace$$\chi_{2107}(1,·)$, $\chi_{2107}(1380,·)$, $\chi_{2107}(2017,·)$, $\chi_{2107}(365,·)$, $\chi_{2107}(1709,·)$, $\chi_{2107}(113,·)$, $\chi_{2107}(1779,·)$, $\chi_{2107}(22,·)$, $\chi_{2107}(484,·)$, $\chi_{2107}(379,·)$, $\chi_{2107}(1212,·)$, $\chi_{2107}(687,·)$, $\chi_{2107}(862,·)$, $\chi_{2107}(127,·)$$\rbrace$
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $\frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{6} - \frac{1}{4}$, $\frac{1}{4} a^{7} - \frac{1}{4} a$, $\frac{1}{8} a^{8} - \frac{1}{8} a^{7} - \frac{1}{8} a^{6} + \frac{3}{8} a^{2} - \frac{3}{8} a - \frac{3}{8}$, $\frac{1}{8} a^{9} - \frac{1}{8} a^{6} - \frac{1}{8} a^{3} + \frac{1}{8}$, $\frac{1}{160} a^{10} - \frac{1}{40} a^{9} + \frac{9}{160} a^{8} - \frac{1}{40} a^{7} + \frac{1}{32} a^{6} + \frac{3}{40} a^{5} + \frac{7}{160} a^{4} + \frac{1}{10} a^{3} + \frac{79}{160} a^{2} - \frac{2}{5} a - \frac{77}{160}$, $\frac{1}{160} a^{11} - \frac{7}{160} a^{9} - \frac{1}{20} a^{8} - \frac{11}{160} a^{7} - \frac{1}{20} a^{6} - \frac{5}{32} a^{5} - \frac{9}{40} a^{4} - \frac{17}{160} a^{3} + \frac{13}{40} a^{2} + \frac{67}{160} a + \frac{13}{40}$, $\frac{1}{3627200} a^{12} - \frac{1013}{3627200} a^{11} - \frac{403}{906800} a^{10} - \frac{98057}{3627200} a^{9} + \frac{10453}{226700} a^{8} + \frac{66103}{725440} a^{7} - \frac{197013}{1813600} a^{6} - \frac{836691}{3627200} a^{5} + \frac{45491}{226700} a^{4} + \frac{585953}{3627200} a^{3} + \frac{184549}{906800} a^{2} - \frac{401819}{3627200} a - \frac{708051}{3627200}$, $\frac{1}{2362694111618943100998641199828770676382997208905976153128120000} a^{13} + \frac{11116339725099809528618202539226945299012680702206722923}{1181347055809471550499320599914385338191498604452988076564060000} a^{12} + \frac{1940150307175342713196597827330119139495372308918446044187091}{2362694111618943100998641199828770676382997208905976153128120000} a^{11} - \frac{1080990197332376479977502262201901818941213956106585738743843}{472538822323788620199728239965754135276599441781195230625624000} a^{10} + \frac{26922968305512539496681534781370978600389072411045599308850919}{472538822323788620199728239965754135276599441781195230625624000} a^{9} - \frac{47531695022629548380011096291286828784517604539991596116844263}{2362694111618943100998641199828770676382997208905976153128120000} a^{8} - \frac{236938326391009630564611596097276646139444800969911825253107211}{2362694111618943100998641199828770676382997208905976153128120000} a^{7} - \frac{5613057903899668876527201950018377521884140354309840230097727}{472538822323788620199728239965754135276599441781195230625624000} a^{6} + \frac{172049955621484664930679041793166468191593684729662791288818737}{2362694111618943100998641199828770676382997208905976153128120000} a^{5} - \frac{552811359000775757097341887751338861356311730578504601521624213}{2362694111618943100998641199828770676382997208905976153128120000} a^{4} - \frac{101508681301975314137214523454532822575071596622356175366159167}{2362694111618943100998641199828770676382997208905976153128120000} a^{3} + \frac{2029464197740827116232375359779800907023312731916723988974853}{8009132581759129155927597287555154835196600708155851366536000} a^{2} + \frac{303262227789568519889773152386217071582276761769996592105437559}{1181347055809471550499320599914385338191498604452988076564060000} a + \frac{14060720922688698818194910776409866269853576061653475418819439}{29907520400239786088590394934541400966873382391214888014280000}$

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

Class group and class number

$C_{40187}$, which has order $40187$ (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:  \( 155488734885.34106 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{14}$ (as 14T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 14
The 14 conjugacy class representatives for $C_{14}$
Character table for $C_{14}$

Intermediate fields

\(\Q(\sqrt{-43}) \), 7.7.87495801462998035849.3

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/3.14.0.1}{14} }$ ${\href{/LocalNumberField/5.2.0.1}{2} }^{7}$ R ${\href{/LocalNumberField/11.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/13.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/17.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/19.14.0.1}{14} }$ ${\href{/LocalNumberField/23.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/29.14.0.1}{14} }$ ${\href{/LocalNumberField/31.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/37.14.0.1}{14} }$ ${\href{/LocalNumberField/41.7.0.1}{7} }^{2}$ R ${\href{/LocalNumberField/47.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/53.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/59.1.0.1}{1} }^{14}$

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
$7$7.14.24.53$x^{14} + 931 x^{13} + 2310 x^{12} + 903 x^{11} + 392 x^{10} + 2198 x^{9} + 2296 x^{8} + 1485 x^{7} + 637 x^{6} + 1295 x^{5} + 2303 x^{4} + 1449 x^{3} + 1316 x^{2} + 2219 x + 2383$$7$$2$$24$$C_{14}$$[2]^{2}$
$43$43.14.13.12$x^{14} + 16659081027$$14$$1$$13$$C_{14}$$[\ ]_{14}$