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Properties

Label 14.0.45288292250...0000.1
Degree $14$
Signature $[0, 7]$
Discriminant $-\,2^{14}\cdot 5^{7}\cdot 29^{12}$
Root discriminant $80.17$
Ramified primes $2, 5, 29$
Class number $3248$ (GRH)
Class group $[2, 2, 2, 406]$ (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![1783541, -244032, 986314, -110934, 240758, -17536, 35841, -2596, 3687, -332, 261, -22, 12, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 2*x^13 + 12*x^12 - 22*x^11 + 261*x^10 - 332*x^9 + 3687*x^8 - 2596*x^7 + 35841*x^6 - 17536*x^5 + 240758*x^4 - 110934*x^3 + 986314*x^2 - 244032*x + 1783541)
 
gp: K = bnfinit(x^14 - 2*x^13 + 12*x^12 - 22*x^11 + 261*x^10 - 332*x^9 + 3687*x^8 - 2596*x^7 + 35841*x^6 - 17536*x^5 + 240758*x^4 - 110934*x^3 + 986314*x^2 - 244032*x + 1783541, 1)
 

Normalized defining polynomial

\( x^{14} - 2 x^{13} + 12 x^{12} - 22 x^{11} + 261 x^{10} - 332 x^{9} + 3687 x^{8} - 2596 x^{7} + 35841 x^{6} - 17536 x^{5} + 240758 x^{4} - 110934 x^{3} + 986314 x^{2} - 244032 x + 1783541 \)

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:  \(-452882922503000372480000000=-\,2^{14}\cdot 5^{7}\cdot 29^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $80.17$
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:  $2, 5, 29$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(580=2^{2}\cdot 5\cdot 29\)
Dirichlet character group:    $\lbrace$$\chi_{580}(1,·)$, $\chi_{580}(139,·)$, $\chi_{580}(161,·)$, $\chi_{580}(459,·)$, $\chi_{580}(141,·)$, $\chi_{580}(239,·)$, $\chi_{580}(401,·)$, $\chi_{580}(81,·)$, $\chi_{580}(339,·)$, $\chi_{580}(181,·)$, $\chi_{580}(59,·)$, $\chi_{580}(281,·)$, $\chi_{580}(219,·)$, $\chi_{580}(199,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{17} a^{12} + \frac{4}{17} a^{11} + \frac{5}{17} a^{10} + \frac{3}{17} a^{9} + \frac{5}{17} a^{8} - \frac{4}{17} a^{7} + \frac{6}{17} a^{6} - \frac{5}{17} a^{5} - \frac{7}{17} a^{4} + \frac{2}{17} a^{3} - \frac{5}{17} a^{2} + \frac{1}{17} a - \frac{1}{17}$, $\frac{1}{21533476451879886599332319755093} a^{13} + \frac{127852920033082372490495925280}{21533476451879886599332319755093} a^{12} - \frac{986379363077531151651847387651}{21533476451879886599332319755093} a^{11} - \frac{9478256870837718648332482318342}{21533476451879886599332319755093} a^{10} - \frac{6339230510180523551627326073450}{21533476451879886599332319755093} a^{9} - \frac{8839625855307234596816284999341}{21533476451879886599332319755093} a^{8} - \frac{6885796957937038652078475389523}{21533476451879886599332319755093} a^{7} - \frac{87860244409477190345149766748}{21533476451879886599332319755093} a^{6} + \frac{6571114414604434278634761849495}{21533476451879886599332319755093} a^{5} + \frac{9411552402581111022908410164120}{21533476451879886599332319755093} a^{4} - \frac{5292222242052558814943942746780}{21533476451879886599332319755093} a^{3} - \frac{8421341446475547652349729466353}{21533476451879886599332319755093} a^{2} - \frac{626839595770420321790230471737}{21533476451879886599332319755093} a + \frac{87265186431773943033452315540}{525206742728777721934934628173}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{406}$, which has order $3248$ (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:  \( 6020.985100147561 \) (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

\(\Q(\sqrt{-5}) \), 7.7.594823321.1

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 R ${\href{/LocalNumberField/3.7.0.1}{7} }^{2}$ R ${\href{/LocalNumberField/7.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/11.14.0.1}{14} }$ ${\href{/LocalNumberField/13.14.0.1}{14} }$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/19.14.0.1}{14} }$ ${\href{/LocalNumberField/23.7.0.1}{7} }^{2}$ R ${\href{/LocalNumberField/31.14.0.1}{14} }$ ${\href{/LocalNumberField/37.14.0.1}{14} }$ ${\href{/LocalNumberField/41.1.0.1}{1} }^{14}$ ${\href{/LocalNumberField/43.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/47.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/53.14.0.1}{14} }$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{7}$

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
$2$2.14.14.15$x^{14} + 2 x^{13} + x^{12} + 4 x^{11} - 2 x^{10} + 2 x^{9} + 4 x^{8} - 2 x^{6} + 4 x^{5} + 4 x^{4} + 2 x^{3} + 4 x^{2} + 1$$2$$7$$14$$C_{14}$$[2]^{7}$
$5$5.14.7.1$x^{14} - 250 x^{8} + 15625 x^{2} - 312500$$2$$7$$7$$C_{14}$$[\ ]_{2}^{7}$
$29$29.7.6.2$x^{7} - 29$$7$$1$$6$$C_7$$[\ ]_{7}$
29.7.6.2$x^{7} - 29$$7$$1$$6$$C_7$$[\ ]_{7}$