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Properties

Label 14.0.47448990293...6087.2
Degree $14$
Signature $[0, 7]$
Discriminant $-\,7^{25}\cdot 29^{12}$
Root discriminant $578.88$
Ramified primes $7, 29$
Class number $29404753$ (GRH)
Class group $[7, 4200679]$ (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![1192124954624, 140686678272, -61788068160, -4870385744, 9763306924, -607090988, -194794943, 14868764, 4778620, 111244, -12586, -812, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 812*x^11 - 12586*x^10 + 111244*x^9 + 4778620*x^8 + 14868764*x^7 - 194794943*x^6 - 607090988*x^5 + 9763306924*x^4 - 4870385744*x^3 - 61788068160*x^2 + 140686678272*x + 1192124954624)
 
gp: K = bnfinit(x^14 - 812*x^11 - 12586*x^10 + 111244*x^9 + 4778620*x^8 + 14868764*x^7 - 194794943*x^6 - 607090988*x^5 + 9763306924*x^4 - 4870385744*x^3 - 61788068160*x^2 + 140686678272*x + 1192124954624, 1)
 

Normalized defining polynomial

\( x^{14} - 812 x^{11} - 12586 x^{10} + 111244 x^{9} + 4778620 x^{8} + 14868764 x^{7} - 194794943 x^{6} - 607090988 x^{5} + 9763306924 x^{4} - 4870385744 x^{3} - 61788068160 x^{2} + 140686678272 x + 1192124954624 \)

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:  \(-474489902930063357496193840307474416087=-\,7^{25}\cdot 29^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $578.88$
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:  $7, 29$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1421=7^{2}\cdot 29\)
Dirichlet character group:    $\lbrace$$\chi_{1421}(1,·)$, $\chi_{1421}(1350,·)$, $\chi_{1421}(778,·)$, $\chi_{1421}(139,·)$, $\chi_{1421}(78,·)$, $\chi_{1421}(1359,·)$, $\chi_{1421}(848,·)$, $\chi_{1421}(1329,·)$, $\chi_{1421}(146,·)$, $\chi_{1421}(20,·)$, $\chi_{1421}(181,·)$, $\chi_{1421}(400,·)$, $\chi_{1421}(1002,·)$, $\chi_{1421}(895,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $\frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{8} a^{4} - \frac{1}{4} a^{3} - \frac{1}{8} a^{2} + \frac{1}{4} a$, $\frac{1}{8} a^{5} - \frac{1}{8} a^{3}$, $\frac{1}{16} a^{6} - \frac{1}{16} a^{5} - \frac{1}{16} a^{4} - \frac{3}{16} a^{3} + \frac{1}{4} a$, $\frac{1}{1856} a^{7} - \frac{1}{64} a^{6} - \frac{1}{64} a^{5} - \frac{3}{64} a^{4} - \frac{1}{16} a^{3} + \frac{1}{16} a^{2} + \frac{1}{4} a$, $\frac{1}{3712} a^{8} - \frac{1}{64} a^{6} - \frac{1}{32} a^{5} + \frac{1}{128} a^{4} - \frac{3}{32} a^{3} + \frac{3}{32} a^{2} - \frac{3}{8} a$, $\frac{1}{3712} a^{9} + \frac{1}{64} a^{6} + \frac{7}{128} a^{5} + \frac{3}{64} a^{4} - \frac{7}{32} a^{3} - \frac{1}{16} a^{2} - \frac{1}{4} a$, $\frac{1}{430592} a^{10} - \frac{1}{14848} a^{9} - \frac{1}{7424} a^{8} - \frac{1}{7424} a^{7} + \frac{349}{14848} a^{6} - \frac{833}{14848} a^{5} + \frac{5}{116} a^{4} + \frac{753}{3712} a^{3} - \frac{5}{32} a^{2} + \frac{1}{4} a$, $\frac{1}{1722368} a^{11} - \frac{1}{861184} a^{10} - \frac{1}{59392} a^{9} + \frac{1}{14848} a^{8} - \frac{1}{59392} a^{7} + \frac{235}{29696} a^{6} + \frac{305}{59392} a^{5} + \frac{43}{3712} a^{4} - \frac{2985}{14848} a^{3} - \frac{5}{64} a^{2} - \frac{15}{32} a + \frac{1}{4}$, $\frac{1}{1987757350912} a^{12} + \frac{3557}{34271678464} a^{11} - \frac{12841}{68543356928} a^{10} + \frac{217085}{4283959808} a^{9} + \frac{3147023}{68543356928} a^{8} - \frac{5123223}{34271678464} a^{7} - \frac{1081951863}{68543356928} a^{6} + \frac{977265953}{17135839232} a^{5} + \frac{20758803}{590891008} a^{4} + \frac{12150549}{147722752} a^{3} + \frac{140955}{1273472} a^{2} - \frac{128997}{318368} a + \frac{1957}{39796}$, $\frac{1}{29469144441420208819122772730287489024} a^{13} + \frac{6871525262115359556778989}{29469144441420208819122772730287489024} a^{12} + \frac{143346657427598011579557195453}{1016177394531731338590440438975430656} a^{11} + \frac{175762991040931070677120470493}{1016177394531731338590440438975430656} a^{10} - \frac{2990348014675495569066274012717}{35040599811439011675532428930187264} a^{9} + \frac{6094805482912752537233122891735}{1016177394531731338590440438975430656} a^{8} - \frac{56041369854953573859201062172489}{1016177394531731338590440438975430656} a^{7} + \frac{15508144895510592115441496915912503}{1016177394531731338590440438975430656} a^{6} + \frac{273525740234675196508420125648275}{31755543579116604330951263717982208} a^{5} + \frac{288772238915075245463520862220453}{8760149952859752918883107232546816} a^{4} - \frac{93062908426254931572381928154699}{2190037488214938229720776808136704} a^{3} - \frac{2790312297168513399473342908623}{18879633519094295083799800070144} a^{2} + \frac{163537743101368550903360445679}{4719908379773573770949950017536} a + \frac{79729870894991153569864200633}{589988547471696721368743752192}$

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

Class group and class number

$C_{7}\times C_{4200679}$, which has order $29404753$ (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:  \( 8981416884.6774 \) (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

\(\Q(\sqrt{-7}) \), 7.7.8233120419813614521.1

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 ${\href{/LocalNumberField/2.1.0.1}{1} }^{14}$ ${\href{/LocalNumberField/3.14.0.1}{14} }$ ${\href{/LocalNumberField/5.14.0.1}{14} }$ R ${\href{/LocalNumberField/11.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/13.14.0.1}{14} }$ ${\href{/LocalNumberField/17.14.0.1}{14} }$ ${\href{/LocalNumberField/19.14.0.1}{14} }$ ${\href{/LocalNumberField/23.7.0.1}{7} }^{2}$ R ${\href{/LocalNumberField/31.14.0.1}{14} }$ ${\href{/LocalNumberField/37.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/41.14.0.1}{14} }$ ${\href{/LocalNumberField/43.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/47.14.0.1}{14} }$ ${\href{/LocalNumberField/53.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/59.14.0.1}{14} }$

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$7.14.25.73$x^{14} + 35 x^{13} + 21 x^{12} + 98 x^{11} - 49 x^{10} + 147 x^{9} - 147 x^{8} + 14 x^{7} - 49 x^{6} + 147 x^{5} + 98 x^{4} + 147 x^{3} + 49 x^{2} - 49 x + 28$$14$$1$$25$$C_{14}$$[2]_{2}$
$29$29.7.6.7$x^{7} - 116$$7$$1$$6$$C_7$$[\ ]_{7}$
29.7.6.7$x^{7} - 116$$7$$1$$6$$C_7$$[\ ]_{7}$