Properties

Label 65.a1
Conductor $65$
Discriminant $65$
j-invariant \( \frac{117649}{65} \)
CM no
Rank $1$
Torsion structure \(\Z/{2}\Z\)

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This is a model for the quotient of the modular curve $X_0(65)$ by its Fricke involution $w_{65}$; this quotient is also denoted $X_0^+(65)$.

Minimal Weierstrass equation

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

\(y^2+xy=x^3-x\)  Toggle raw display

Mordell-Weil group structure

$\Z\times \Z/{2}\Z$

Infinite order Mordell-Weil generator and height

sage: E.gens()
 
magma: Generators(E);
 

$P$ =  \(\left(1, 0\right)\)  Toggle raw display
$\hat{h}(P)$ ≈  $0.37551409866126632180447287682$

Torsion generators

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

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

Integral points

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

\( \left(-1, 1\right) \), \( \left(-1, 0\right) \), \( \left(0, 0\right) \), \( \left(1, 0\right) \), \( \left(1, -1\right) \), \( \left(4, 6\right) \), \( \left(4, -10\right) \)  Toggle raw display

Invariants

sage: E.conductor().factor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor: \( 65 \)  =  $5 \cdot 13$
sage: E.discriminant().factor()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant: $65 $  =  $5 \cdot 13 $
sage: E.j_invariant().factor()
 
gp: E.j
 
magma: jInvariant(E);
 
j-invariant: \( \frac{117649}{65} \)  =  $5^{-1} \cdot 7^{6} \cdot 13^{-1}$
Endomorphism ring: $\Z$
Geometric endomorphism ring: \(\Z\) (no potential complex multiplication)
Sato-Tate group: $\mathrm{SU}(2)$
Faltings height: $-0.96161523729216172163187933912\dots$
Stable Faltings height: $-0.96161523729216172163187933912\dots$

BSD invariants

sage: E.rank()
 
magma: Rank(E);
 
Analytic rank: $1$
sage: E.regulator()
 
magma: Regulator(E);
 
Regulator: $0.37551409866126632180447287682\dots$
sage: E.period_lattice().omega()
 
gp: E.omega[1]
 
magma: RealPeriod(E);
 
Real period: $5.3828534705718009941152235294\dots$
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: $ 1 $
sage: E.torsion_order()
 
gp: elltors(E)[1]
 
magma: Order(TorsionSubgroup(E));
 
Torsion order: $2$
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) $ ≈ $ 0.50533434230685977745953442460952885255 $

Modular invariants

Modular form   65.2.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);
 

\( q - q^{2} - 2q^{3} - q^{4} - q^{5} + 2q^{6} - 4q^{7} + 3q^{8} + q^{9} + q^{10} + 2q^{11} + 2q^{12} - q^{13} + 4q^{14} + 2q^{15} - q^{16} + 2q^{17} - q^{18} - 6q^{19} + O(q^{20}) \)  Toggle raw display

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

sage: E.modular_degree()
 
magma: ModularDegree(E);
 
Modular degree: 2
$ \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$(j)_{-}$
$5$ $1$ $I_{1}$ Non-split multiplicative 1 1 1 1
$13$ $1$ $I_{1}$ Non-split multiplicative 1 1 1 1

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 for all primes $\ell$ except those listed in the table below.

prime $\ell$ mod-$\ell$ image $\ell$-adic image
$2$ 2B 8.12.0.22

$p$-adic regulators

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

Note: $p$-adic regulator data only exists for primes $p\ge 5$ of good ordinary reduction.

Iwasawa invariants

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47
Reduction type ordinary ordinary nonsplit ordinary ordinary nonsplit ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary
$\lambda$-invariant(s) 2 1 1 1 1 1 1 1 1 1 1 1 1 3 1
$\mu$-invariant(s) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

Isogenies

This curve has non-trivial cyclic isogenies of degree $d$ for $d=$ 2.
Its isogeny class 65.a consists of 2 curves linked by isogenies of degree 2.

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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z$ are as follows:

$[K:\Q]$ $K$ $E(K)_{\rm tors}$ Base change curve
$2$ \(\Q(\sqrt{65}) \) \(\Z/2\Z \times \Z/2\Z\) 2.2.65.1-65.1-a2
$4$ 4.0.1040.2 \(\Z/4\Z\) Not in database
$8$ 8.4.1206702250000.5 \(\Z/2\Z \times \Z/4\Z\) Not in database
$8$ 8.0.4569760000.3 \(\Z/2\Z \times \Z/4\Z\) Not in database
$8$ 8.2.39039316875.1 \(\Z/6\Z\) Not in database
$16$ Deg 16 \(\Z/4\Z \times \Z/4\Z\) Not in database
$16$ Deg 16 \(\Z/8\Z\) Not in database
$16$ Deg 16 \(\Z/2\Z \times \Z/6\Z\) 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 $E$, together with the $2$-isogenous curve [65.a2 ], have the minimal conductor for elliptic curves over $\bf Q$ whose Mordell-Weil group have positive rank and nontrivial torsion. This is related to the identification of $E$ with the quotient of the modular curve $X_0(65)$ by its Fricke involution $w_{65}$: the Atkin-Lehner involutions $w_5$ and $w_{13}$ descend to an involution of $E$ that has no fixed points (because each of $5$ and $13$ is a quadratic nonresidue of the other) and thus must be translation by a rational $2$-torsion point on $E$.