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Properties

Label 14.0.53588606915...5607.1
Degree $14$
Signature $[0, 7]$
Discriminant $-\,7^{25}\cdot 43^{12}$
Root discriminant $811.37$
Ramified primes $7, 43$
Class number $1196776$ (GRH)
Class group $[2, 14, 42742]$ (GRH)
Galois group $C_{14}$ (as 14T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![657479876713, -2009656697825, 2062378028046, 94982321987, -3402184037, 2524506264, 210375522, 7961708, 13842689, -68026, 301, -7224, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 7224*x^11 + 301*x^10 - 68026*x^9 + 13842689*x^8 + 7961708*x^7 + 210375522*x^6 + 2524506264*x^5 - 3402184037*x^4 + 94982321987*x^3 + 2062378028046*x^2 - 2009656697825*x + 657479876713)
 
gp: K = bnfinit(x^14 - 7224*x^11 + 301*x^10 - 68026*x^9 + 13842689*x^8 + 7961708*x^7 + 210375522*x^6 + 2524506264*x^5 - 3402184037*x^4 + 94982321987*x^3 + 2062378028046*x^2 - 2009656697825*x + 657479876713, 1)
 

Normalized defining polynomial

\( x^{14} - 7224 x^{11} + 301 x^{10} - 68026 x^{9} + 13842689 x^{8} + 7961708 x^{7} + 210375522 x^{6} + 2524506264 x^{5} - 3402184037 x^{4} + 94982321987 x^{3} + 2062378028046 x^{2} - 2009656697825 x + 657479876713 \)

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:  \(-53588606915566584613059849086223224055607=-\,7^{25}\cdot 43^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $811.37$
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$$\chi_{2107}(2080,·)$, $\chi_{2107}(1,·)$, $\chi_{2107}(1602,·)$, $\chi_{2107}(643,·)$, $\chi_{2107}(580,·)$, $\chi_{2107}(993,·)$, $\chi_{2107}(1420,·)$, $\chi_{2107}(1387,·)$, $\chi_{2107}(1196,·)$, $\chi_{2107}(78,·)$, $\chi_{2107}(1693,·)$, $\chi_{2107}(1870,·)$, $\chi_{2107}(729,·)$, $\chi_{2107}(477,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{473} a^{7} - \frac{2}{11} a^{6} + \frac{5}{11} a^{5} - \frac{4}{11} a^{3} - \frac{4}{11} a^{2} + \frac{5}{11} a$, $\frac{1}{473} a^{8} - \frac{2}{11} a^{6} + \frac{1}{11} a^{5} - \frac{4}{11} a^{4} + \frac{4}{11} a^{3} + \frac{2}{11} a^{2} + \frac{1}{11} a$, $\frac{1}{473} a^{9} + \frac{5}{11} a^{6} - \frac{3}{11} a^{5} + \frac{4}{11} a^{4} - \frac{1}{11} a^{3} - \frac{2}{11} a^{2} + \frac{1}{11} a$, $\frac{1}{20339} a^{10} - \frac{79}{473} a^{6} + \frac{95}{473} a^{5} + \frac{197}{473} a^{4} - \frac{8}{43} a^{3} - \frac{3}{11} a^{2} - \frac{3}{11} a$, $\frac{1}{223729} a^{11} - \frac{1}{223729} a^{10} - \frac{4}{5203} a^{9} + \frac{1}{5203} a^{8} - \frac{3}{5203} a^{7} - \frac{1159}{5203} a^{6} - \frac{1919}{5203} a^{5} - \frac{2564}{5203} a^{4} + \frac{1421}{5203} a^{3} + \frac{58}{121} a^{2} - \frac{5}{121} a - \frac{5}{11}$, $\frac{1}{1806082376260613} a^{12} - \frac{21101230}{42001915726991} a^{11} - \frac{918351969}{42001915726991} a^{10} - \frac{27638281988}{42001915726991} a^{9} + \frac{41877101447}{42001915726991} a^{8} - \frac{19987628458}{42001915726991} a^{7} + \frac{16146678880240}{42001915726991} a^{6} + \frac{19248851481709}{42001915726991} a^{5} - \frac{277230530813}{976788737837} a^{4} + \frac{46583842270}{976788737837} a^{3} + \frac{3978149849}{22716017159} a^{2} - \frac{6077006808}{22716017159} a + \frac{956183928}{2065092469}$, $\frac{1}{2231414553123600784114463154181944338908157893813} a^{13} - \frac{256717207605972593482653999408228}{2231414553123600784114463154181944338908157893813} a^{12} - \frac{1987478867681746312282988523839621040834}{4717578336413532313138399903133074712279403581} a^{11} + \frac{321529200621238812829691794115609298077643}{51893361700548855444522398934463821835073439391} a^{10} - \frac{14096623143060132105026202318704415820165475}{51893361700548855444522398934463821835073439391} a^{9} - \frac{2629861221763815394180044745445456956258426}{51893361700548855444522398934463821835073439391} a^{8} + \frac{44327002606147278395239602082085930985813398}{51893361700548855444522398934463821835073439391} a^{7} - \frac{2158889065658815009988555804605325987901605758}{4717578336413532313138399903133074712279403581} a^{6} + \frac{22776417349377409472109400583183873763659608101}{51893361700548855444522398934463821835073439391} a^{5} - \frac{312624908376258483089323796366012786996896372}{1206822365129043149872613928708460972908684637} a^{4} + \frac{593358715335086696668042584242168392847162734}{1206822365129043149872613928708460972908684637} a^{3} + \frac{3662723082491179758188135603387078911786399}{28065636398349840694711951830429324951364759} a^{2} + \frac{9138040594884052130290692058539552991265031}{28065636398349840694711951830429324951364759} a - \frac{184830034759917120189410121342376130124132}{2551421490759076426791995620948120450124069}$

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

Class group and class number

$C_{2}\times C_{14}\times C_{42742}$, which has order $1196776$ (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:  \( 769768169.7365845 \) (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

\(\Q(\sqrt{-7}) \), 7.7.87495801462998035849.6

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 ${\href{/LocalNumberField/2.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/3.14.0.1}{14} }$ ${\href{/LocalNumberField/5.14.0.1}{14} }$ R ${\href{/LocalNumberField/11.1.0.1}{1} }^{14}$ ${\href{/LocalNumberField/13.14.0.1}{14} }$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/19.14.0.1}{14} }$ ${\href{/LocalNumberField/23.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/29.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/31.14.0.1}{14} }$ ${\href{/LocalNumberField/37.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{7}$ R ${\href{/LocalNumberField/47.14.0.1}{14} }$ ${\href{/LocalNumberField/53.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/59.14.0.1}{14} }$

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
$7$7.14.25.59$x^{14} - 168 x^{13} + 70 x^{12} - 147 x^{11} + 147 x^{10} - 98 x^{9} + 49 x^{8} + 168 x^{7} - 49 x^{4} - 147 x^{3} - 49 x^{2} + 98 x + 126$$14$$1$$25$$C_{14}$$[2]_{2}$
$43$43.7.6.4$x^{7} + 31347$$7$$1$$6$$C_7$$[\ ]_{7}$
43.7.6.4$x^{7} + 31347$$7$$1$$6$$C_7$$[\ ]_{7}$