Properties

Label 14.a5
Conductor $14$
Discriminant $-28$
j-invariant \( -\frac{15625}{28} \)
CM no
Rank $0$
Torsion structure \(\Z/{6}\Z\)

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This is a model for the modular curve $X_1(14)$.

Minimal Weierstrass equation

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

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

Mordell-Weil group structure

\(\Z/{6}\Z\)

Torsion generators

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

\( \left(1, 0\right) \)  Toggle raw display

Integral points

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

\( \left(-1, 0\right) \), \( \left(0, 0\right) \), \( \left(0, -1\right) \), \( \left(1, 0\right) \), \( \left(1, -2\right) \)  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: \(-28 \)  =  \(-1 \cdot 2^{2} \cdot 7 \)
sage: E.j_invariant().factor()
 
gp: E.j
 
magma: jInvariant(E);
 
j-invariant: \( -\frac{15625}{28} \)  =  \(-1 \cdot 2^{-2} \cdot 5^{6} \cdot 7^{-1}\)
Endomorphism ring: \(\Z\)
Geometric endomorphism ring: \(\Z\) (no potential complex multiplication)
Sato-Tate group: $\mathrm{SU}(2)$
Faltings height: \(-1.0320848985249978885230789428\dots\)
Stable Faltings height: \(-1.0320848985249978885230789428\dots\)

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: \(5.9440258682006497025087150302\dots\)
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: \( 2 \)  = \( 2\cdot1 \)
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)

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: 3
\( \Gamma_0(N) \)-optimal: no
Manin constant: 3

Special L-value

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);
 

\( L(E,1) \) ≈ \( 0.33022365934448053902826194612222809986 \)

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\((j)_{-}\)
\(2\) \(2\) \(I_{2}\) Non-split multiplicative 1 1 2 2
\(7\) \(1\) \(I_{1}\) Split multiplicative -1 1 1 1

Galois representations

The image of the 2-adic representation attached to this elliptic curve is the subgroup of $\GL(2,\Z_2)$ with Rouse label X16.

This subgroup is the pull-back of the subgroup of $\GL(2,\Z_2/2^3\Z_2)$ generated by $\left(\begin{array}{rr} 1 & 1 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 7 & 0 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 5 & 0 \\ 2 & 1 \end{array}\right)$ and has index 6.

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 mod \( p \) Galois representation has maximal image \(\GL(2,\F_p)\) for all primes \( p \) except those listed.

prime Image of Galois representation
\(2\) B
\(3\) B.1.1

$p$-adic data

$p$-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 0 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, 6, 9 and 18.
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-a6
$3$ \(\Q(\zeta_{7})^+\) \(\Z/18\Z\) 3.3.49.1-56.1-a5
$4$ 4.2.448.1 \(\Z/12\Z\) Not in database
$6$ 6.0.1037232.1 \(\Z/3\Z \times \Z/6\Z\) Not in database
$6$ 6.0.21168.1 \(\Z/18\Z\) Not in database
$6$ \(\Q(\zeta_{7})\) \(\Z/2\Z \times \Z/18\Z\) Not in database
$8$ 8.0.9834496.2 \(\Z/2\Z \times \Z/12\Z\) Not in database
$8$ 8.0.120472576.1 \(\Z/2\Z \times \Z/12\Z\) Not in database
$12$ 12.0.52716660869376.1 \(\Z/6\Z \times \Z/6\Z\) Not in database
$12$ 12.0.1075850221824.1 \(\Z/2\Z \times \Z/18\Z\) Not in database
$12$ 12.6.10578455953408.1 \(\Z/36\Z\) Not in database
$16$ Deg 16 \(\Z/24\Z\) Not in database
$18$ 18.0.1115906277282951168.1 \(\Z/3\Z \times \Z/18\Z\) 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.

Additional information

This curve $E$ also parametrizes pairs $(R,T)$ where $R$ is a rational rectangle, $T$ is a Pythagorean triangle, and $R,T$ have the same perimeter and the same area. (That is, $E$ is birational with the curve of $(a:b:c:x:y) \in {\bf P}^4$ with $a^2+b^2=c^2$, $a+b+c=2x+2y$, and $ab/2=xy$.) Unfortunately the six rational points on $E$ all yield degenerate solutions. [Noted in passing in Richard Guy's paper "My Favorite Elliptic Curve: A Tale of Two Types of Triangles", Amer. Math. Monthly 102 #9 (Nov. 1995), 771-781.]