↑

Properties

Label 14.a6
Conductor $14$
Discriminant $-21952$
j-invariant \( \frac{9938375}{21952} \)
CM no
Rank $0$
Torsion structure \(\Z/{6}\Z\)

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath

This is a model for the modular curve $X_0(14)$.

Minimal Weierstrass equation

sage: E = EllipticCurve([1, 0, 1, 4, -6])
 
gp: E = ellinit([1, 0, 1, 4, -6])
 
magma: E := EllipticCurve([1, 0, 1, 4, -6]);
 

\(y^2+xy+y=x^3+4x-6\)  Toggle raw display

Mordell-Weil group structure

$\Z/{6}\Z$

Torsion generators

sage: E.torsion_subgroup().gens()
 
gp: elltors(E)
 
magma: TorsionSubgroup(E);
 

\( \left(9, 23\right) \)  Toggle raw display

Integral points

sage: E.integral_points()
 
magma: IntegralPoints(E);
 

\( \left(1, -1\right) \), \( \left(2, 2\right) \), \( \left(2, -5\right) \), \( \left(9, 23\right) \), \( \left(9, -33\right) \)  Toggle raw display

Invariants

sage: E.conductor().factor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor: \( 14 \)  =  $2 \cdot 7$
sage: E.discriminant().factor()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant: $-21952 $  =  $-1 \cdot 2^{6} \cdot 7^{3} $
sage: E.j_invariant().factor()
 
gp: E.j
 
magma: jInvariant(E);
 
j-invariant: \( \frac{9938375}{21952} \)  =  $2^{-6} \cdot 5^{3} \cdot 7^{-3} \cdot 43^{3}$
Endomorphism ring: $\Z$
Geometric endomorphism ring: \(\Z\) (no potential complex multiplication)
Sato-Tate group: $\mathrm{SU}(2)$
Faltings height: $-0.48277875419094304282545632435\dots$
Stable Faltings height: $-0.48277875419094304282545632435\dots$

BSD invariants

sage: E.rank()
 
magma: Rank(E);
 
Analytic rank: $0$
sage: E.regulator()
 
magma: Regulator(E);
 
Regulator: $1$
sage: E.period_lattice().omega()
 
gp: E.omega[1]
 
magma: RealPeriod(E);
 
Real period: $1.9813419560668832341695716767\dots$
sage: E.tamagawa_numbers()
 
gp: gr=ellglobalred(E); [[gr[4][i,1],gr[5][i][4]] | i<-[1..#gr[4][,1]]]
 
magma: TamagawaNumbers(E);
 
Tamagawa product: $ 6 $  = $ 2\cdot3 $
sage: E.torsion_order()
 
gp: elltors(E)[1]
 
magma: Order(TorsionSubgroup(E));
 
Torsion order: $6$
sage: E.sha().an_numerical()
 
magma: MordellWeilShaInformation(E);
 
Analytic order of Ш: $1$ (exact)
sage: r = E.rank();
 
sage: E.lseries().dokchitser().derivative(1,r)/r.factorial()
 
gp: ar = ellanalyticrank(E);
 
gp: ar[2]/factorial(ar[1])
 
magma: Lr1 where r,Lr1 := AnalyticRank(E: Precision:=12);
 
Special value: $ L(E,1) $ ≈ $ 0.33022365934448053902826194612222809986 $

Modular invariants

Modular form   14.2.a.a

sage: E.q_eigenform(20)
 
gp: xy = elltaniyama(E);
 
gp: x*deriv(xy[1])/(2*xy[2]+E.a1*xy[1]+E.a3)
 
magma: ModularForm(E);
 

\( q - q^{2} - 2q^{3} + q^{4} + 2q^{6} + q^{7} - q^{8} + q^{9} - 2q^{12} - 4q^{13} - q^{14} + q^{16} + 6q^{17} - q^{18} + 2q^{19} + O(q^{20}) \)  Toggle raw display

For more coefficients, see the Downloads section to the right.

sage: E.modular_degree()
 
magma: ModularDegree(E);
 
Modular degree: 1
$ \Gamma_0(N) $-optimal: yes
Manin constant: 1

Local data

This elliptic curve is semistable. There are 2 primes of bad reduction:

sage: E.local_data()
 
gp: ellglobalred(E)[5]
 
magma: [LocalInformation(E,p) : p in BadPrimes(E)];
 
prime Tamagawa number Kodaira symbol Reduction type Root number ord($N$) ord($\Delta$) ord$(j)_{-}$
$2$ $2$ $I_{6}$ Non-split multiplicative 1 1 6 6
$7$ $3$ $I_{3}$ Split multiplicative -1 1 3 3

Galois representations

sage: rho = E.galois_representation();
 
sage: [rho.image_type(p) for p in rho.non_surjective()]
 
magma: [GaloisRepresentation(E,p): p in PrimesUpTo(20)];
 

The $\ell$-adic Galois representation has maximal image $\GL(2,\Z_\ell)$ for all primes $\ell$ except those listed in the table below.

prime $\ell$ mod-$\ell$ image $\ell$-adic image
$2$ 2B 8.6.0.1
$3$ 3Cs.1.1 3.24.0.1

$p$-adic regulators

sage: [E.padic_regulator(p) for p in primes(5,20) if E.conductor().valuation(p)<2]
 

All $p$-adic regulators are identically $1$ since the rank is $0$.

Iwasawa invariants

$p$ 2 3 7
Reduction type nonsplit ordinary split
$\lambda$-invariant(s) 0 0 1
$\mu$-invariant(s) 0 1 0

All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 5$ of good reduction are zero.

Isogenies

This curve has non-trivial cyclic isogenies of degree $d$ for $d=$ 2, 3 and 6.
Its isogeny class 14.a consists of 6 curves linked by isogenies of degrees dividing 18.

Growth of torsion in number fields

The number fields $K$ of degree less than 24 such that $E(K)_{\rm tors}$ is strictly larger than $E(\Q)_{\rm tors}$ $\cong \Z/{6}\Z$ are as follows:

$[K:\Q]$ $K$ $E(K)_{\rm tors}$ Base change curve
$2$ \(\Q(\sqrt{-7}) \) \(\Z/2\Z \times \Z/6\Z\) 2.0.7.1-28.2-a7
$2$ \(\Q(\sqrt{-3}) \) \(\Z/3\Z \times \Z/6\Z\) 2.0.3.1-196.2-a3
$4$ 4.2.448.1 \(\Z/12\Z\) Not in database
$4$ \(\Q(\sqrt{-3}, \sqrt{-7})\) \(\Z/6\Z \times \Z/6\Z\) Not in database
$8$ 8.0.120472576.1 \(\Z/2\Z \times \Z/12\Z\) Not in database
$8$ 8.0.9834496.2 \(\Z/2\Z \times \Z/12\Z\) Not in database
$8$ 8.0.16257024.1 \(\Z/3\Z \times \Z/12\Z\) Not in database
$9$ 9.3.2315685267.2 \(\Z/18\Z\) Not in database
$16$ Deg 16 \(\Z/24\Z\) Not in database
$16$ Deg 16 \(\Z/6\Z \times \Z/12\Z\) Not in database
$16$ 16.0.634562281237118976.5 \(\Z/6\Z \times \Z/12\Z\) Not in database
$18$ 18.0.144784752906623254803.2 \(\Z/3\Z \times \Z/18\Z\) Not in database
$18$ 18.0.3674701490222020866048.1 \(\Z/3\Z \times \Z/18\Z\) Not in database
$18$ 18.0.432324955623130532869681152.2 \(\Z/3\Z \times \Z/18\Z\) Not in database
$18$ 18.0.1839302601739695422127.2 \(\Z/2\Z \times \Z/18\Z\) Not in database

We only show fields where the torsion growth is primitive. For fields not in the database, click on the degree shown to reveal the defining polynomial.