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Properties

Label 14.0.53588606915...5607.6
Degree $14$
Signature $[0, 7]$
Discriminant $-\,7^{25}\cdot 43^{12}$
Root discriminant $811.37$
Ramified primes $7, 43$
Class number $31709713$ (GRH)
Class group $[7, 4529959]$ (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![89621816934400, -18134414520320, -330830535168, -104675735920, 89548734100, -6638516472, 292188225, -137092600, 13355972, -156520, 57190, -7224, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 7224*x^11 + 57190*x^10 - 156520*x^9 + 13355972*x^8 - 137092600*x^7 + 292188225*x^6 - 6638516472*x^5 + 89548734100*x^4 - 104675735920*x^3 - 330830535168*x^2 - 18134414520320*x + 89621816934400)
 
gp: K = bnfinit(x^14 - 7224*x^11 + 57190*x^10 - 156520*x^9 + 13355972*x^8 - 137092600*x^7 + 292188225*x^6 - 6638516472*x^5 + 89548734100*x^4 - 104675735920*x^3 - 330830535168*x^2 - 18134414520320*x + 89621816934400, 1)
 

Normalized defining polynomial

\( x^{14} - 7224 x^{11} + 57190 x^{10} - 156520 x^{9} + 13355972 x^{8} - 137092600 x^{7} + 292188225 x^{6} - 6638516472 x^{5} + 89548734100 x^{4} - 104675735920 x^{3} - 330830535168 x^{2} - 18134414520320 x + 89621816934400 \)

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$$\chi_{2107}(1,·)$, $\chi_{2107}(895,·)$, $\chi_{2107}(484,·)$, $\chi_{2107}(2085,·)$, $\chi_{2107}(1994,·)$, $\chi_{2107}(1420,·)$, $\chi_{2107}(365,·)$, $\chi_{2107}(398,·)$, $\chi_{2107}(1779,·)$, $\chi_{2107}(1380,·)$, $\chi_{2107}(90,·)$, $\chi_{2107}(379,·)$, $\chi_{2107}(1245,·)$, $\chi_{2107}(127,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $\frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{8} a^{4} - \frac{1}{4} a^{3} - \frac{1}{8} a^{2} + \frac{1}{4} a$, $\frac{1}{8} a^{5} - \frac{1}{8} a^{3}$, $\frac{1}{16} a^{6} - \frac{1}{16} a^{5} - \frac{1}{16} a^{4} - \frac{3}{16} a^{3} + \frac{1}{4} a$, $\frac{1}{2752} a^{7} - \frac{1}{32} a^{6} - \frac{1}{32} a^{5} + \frac{7}{64} a^{3} + \frac{1}{32} a^{2} - \frac{1}{8} a$, $\frac{1}{5504} a^{8} - \frac{1}{64} a^{6} + \frac{3}{128} a^{4} - \frac{1}{8} a^{3} - \frac{1}{32} a^{2} + \frac{1}{8} a$, $\frac{1}{11008} a^{9} - \frac{1}{11008} a^{8} - \frac{1}{5504} a^{7} - \frac{3}{128} a^{6} + \frac{11}{256} a^{5} - \frac{3}{256} a^{4} - \frac{7}{32} a^{3} + \frac{3}{64} a^{2} + \frac{3}{16} a$, $\frac{1}{2366720} a^{10} - \frac{1}{11008} a^{8} - \frac{1}{13760} a^{7} + \frac{137}{11008} a^{6} + \frac{81}{1376} a^{5} + \frac{2681}{55040} a^{4} + \frac{335}{2752} a^{3} + \frac{1}{64} a^{2} - \frac{39}{80} a$, $\frac{1}{75735040} a^{11} - \frac{9}{352256} a^{9} - \frac{3}{220160} a^{8} + \frac{17}{352256} a^{7} - \frac{49}{5504} a^{6} - \frac{57519}{1761280} a^{5} + \frac{1221}{44032} a^{4} - \frac{217}{2048} a^{3} - \frac{69}{2560} a^{2} - \frac{11}{32} a$, $\frac{1}{6513213440} a^{12} - \frac{1}{151470080} a^{11} - \frac{13}{151470080} a^{10} + \frac{3831}{151470080} a^{9} - \frac{3}{151470080} a^{8} + \frac{2681}{151470080} a^{7} + \frac{3870961}{151470080} a^{6} + \frac{8493997}{151470080} a^{5} - \frac{54087}{880640} a^{4} - \frac{121621}{880640} a^{3} + \frac{109}{5120} a^{2} + \frac{117}{320} a$, $\frac{1}{261949987878963832879829160936238639022080} a^{13} - \frac{3328910228060789212134503949417}{65487496969740958219957290234059659755520} a^{12} + \frac{849025791270463784360067544873}{1522965045807929260929239307768829296640} a^{11} - \frac{291901134888953990191198371677289}{1522965045807929260929239307768829296640} a^{10} - \frac{90397195616067586772915450596564229}{3045930091615858521858478615537658593280} a^{9} - \frac{134963019592384923087698619404967271}{1522965045807929260929239307768829296640} a^{8} - \frac{24126349799649329445021733185586571}{152296504580792926092923930776882929664} a^{7} - \frac{5679418477129333454254373763781388767}{1522965045807929260929239307768829296640} a^{6} - \frac{91752829017383804983724786641661686929}{6091860183231717043716957231075317186560} a^{5} + \frac{316259402750005982161044521633055773}{17708895881487549545688829160102666240} a^{4} - \frac{2915805643415669272271549988393816287}{35417791762975099091377658320205332480} a^{3} + \frac{14980334612002886002786596005182147}{205917393970785459833591036745379840} a^{2} - \frac{4215300781858069359510814633181377}{12869837123174091239599439796586240} a - \frac{3571579204526600892859222772623}{10054560252479758780937062341083}$

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

Class group and class number

$C_{7}\times C_{4529959}$, which has order $31709713$ (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

\(\Q(\sqrt{-7}) \), 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 ${\href{/LocalNumberField/2.1.0.1}{1} }^{14}$ ${\href{/LocalNumberField/3.14.0.1}{14} }$ ${\href{/LocalNumberField/5.2.0.1}{2} }^{7}$ R ${\href{/LocalNumberField/11.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/13.14.0.1}{14} }$ ${\href{/LocalNumberField/17.14.0.1}{14} }$ ${\href{/LocalNumberField/19.14.0.1}{14} }$ ${\href{/LocalNumberField/23.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/29.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/31.14.0.1}{14} }$ ${\href{/LocalNumberField/37.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/41.14.0.1}{14} }$ R ${\href{/LocalNumberField/47.14.0.1}{14} }$ ${\href{/LocalNumberField/53.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{7}$

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}$$[2]_{2}$
$43$43.7.6.3$x^{7} - 3483$$7$$1$$6$$C_7$$[\ ]_{7}$
43.7.6.3$x^{7} - 3483$$7$$1$$6$$C_7$$[\ ]_{7}$