Properties

Label 37.a1
Conductor $37$
Discriminant $37$
j-invariant \( \frac{110592}{37} \)
CM no
Rank $1$
Torsion structure trivial

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This is the elliptic curve of minimal conductor with positive rank. It is also a model for the quotient of the modular curve $X_0(37)$ by its Fricke involution $w_{37}$; this quotient is also denoted $X_0^+(37)$. This is the smallest prime $N \in \mathbb{N}$ such that $X_0(N)/ \langle w_N \rangle$ is of positive genus.

Minimal Weierstrass equation

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

Minimal equation

Minimal equation

Simplified equation

\(y^2+y=x^3-x\) Copy content Toggle raw display (homogenize, simplify)
\(y^2z+yz^2=x^3-xz^2\) Copy content Toggle raw display (dehomogenize, simplify)
\(y^2=x^3-16x+16\) Copy content Toggle raw display (homogenize, minimize)

Mordell-Weil group structure

\(\Z\)

Infinite order Mordell-Weil generator and height

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

$P$ =  \(\left(0, 0\right)\) Copy content Toggle raw display
$\hat{h}(P)$ ≈  $0.051111408239968840235886099757$

Integral points

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

\( \left(-1, 0\right) \), \( \left(-1, -1\right) \), \( \left(0, 0\right) \), \( \left(0, -1\right) \), \( \left(1, 0\right) \), \( \left(1, -1\right) \), \( \left(2, 2\right) \), \( \left(2, -3\right) \), \( \left(6, 14\right) \), \( \left(6, -15\right) \) Copy content Toggle raw display

Invariants

sage: E.conductor().factor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor: \( 37 \)  =  $37$
sage: E.discriminant().factor()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant: $37 $  =  $37 $
sage: E.j_invariant().factor()
 
gp: E.j
 
magma: jInvariant(E);
 
j-invariant: \( \frac{110592}{37} \)  =  $2^{12} \cdot 3^{3} \cdot 37^{-1}$
Endomorphism ring: $\Z$
Geometric endomorphism ring: \(\Z\) (no potential complex multiplication)
Sato-Tate group: $\mathrm{SU}(2)$
Faltings height: $-0.99654220763736714794656344357\dots$
Stable Faltings height: $-0.99654220763736714794656344357\dots$

BSD invariants

sage: E.rank()
 
magma: Rank(E);
 
Analytic rank: $1$
sage: E.regulator()
 
magma: Regulator(E);
 
Regulator: $0.051111408239968840235886099757\dots$
sage: E.period_lattice().omega()
 
gp: E.omega[1]
 
magma: RealPeriod(E);
 
Real period: $5.9869172924639192596640199589\dots$
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: $1$
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) $ ≈ $ 0.30599977383405230182048368332 $

Modular invariants

Modular form   37.2.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);
 

\( q - 2 q^{2} - 3 q^{3} + 2 q^{4} - 2 q^{5} + 6 q^{6} - q^{7} + 6 q^{9} + 4 q^{10} - 5 q^{11} - 6 q^{12} - 2 q^{13} + 2 q^{14} + 6 q^{15} - 4 q^{16} - 12 q^{18} + O(q^{20}) \) Copy content 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 is only one prime 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$(j)_{-}$
$37$ $1$ $I_{1}$ 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 $\ell$.

$p$-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 ss ss ord ord ord ord ss ss ord ord ord nonsplit ord ord ord
$\lambda$-invariant(s) 2,1 1,5 1 1 1 3 1,3 1,1 1 1 1 1 1 1 1
$\mu$-invariant(s) 0,0 0,0 0 0 0 0 0,0 0,0 0 0 0 0 0 0 0

Isogenies

This curve has no rational isogenies. Its isogeny class 37.a consists of this curve only.

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 $E(\Q)_{\rm tors}$ (which is trivial) are as follows:

$[K:\Q]$ $K$ $E(K)_{\rm tors}$ Base change curve
$3$ 3.3.148.1 \(\Z/2\Z\) Not in database
$6$ 6.6.810448.1 \(\Z/2\Z \oplus \Z/2\Z\) Not in database
$8$ 8.2.4098790107.1 \(\Z/3\Z\) Not in database
$12$ Deg 12 \(\Z/4\Z\) 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 elliptic curve $E$ has the least conductor of any elliptic curve over $\Q$ that is the sole member of its isogeny class.

This elliptic curve is associated to the [Somos-4 sequence ] $\{a(n)\}$. Let $P$ be the generator $(0,0)$ of $E(\Q)$. Then for odd $n$ the $x$- and $y$-coordinates of $nP+T$ have denominators $d_n^2$ and $d_n^3$ where $$d_n = 1, 1, 2, 3, 7, 23, 59, 314, 1529, 8209$$ for $n=1,3,5,\ldots,19$, and $d_{2n-3} = a(n)$ in general, satisfying the Somos-4 recurrence $$ d_n d_{n+4} = d_{n+1} d_{n+3} + d_{n+2}^2. $$ The regulator of $E$, which is equal to the canonical height $\hat h(P) \approx 0.0511$, controls the growth of the $a(n)$: asymptotically $\log a_n \sim 2 \hat h(P) n^2$.

The integral points on $E: y^2+y=x^3-x$ correspond to solutions of the classical problem of finding all integers that are simultaneously the product of two consecutive integers and the product of three consecutive integers [since $y^2+y=y(y+1)$ and $x^3-x = (x-1)x(x+1)$]. The fact that $210 = 5 \cdot 6 \cdot 7 = 14 \cdot 15$ is the last such example can be proved easily from the fact that $(0,0)$ generates the group of rational solutions. See J.H.Silverman, The Arithmetic of Elliptic Curves (Springer GTM 106, 1985), page 275, exercise 9.13 [10.1007/978-0-387-09494-6 ].