# Properties

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

# Related objects

Show commands for: Magma / Pari/GP / SageMath

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 is the smallest prime $N \in \mathbb{N}$ such that $X_0(N)/w_N$ is of positive genus.

## Minimal Weierstrass equation

sage: E = EllipticCurve([0, 0, 1, -1, 0]) # or

sage: E = EllipticCurve("37a1")

gp: E = ellinit([0, 0, 1, -1, 0]) \\ or

gp: E = ellinit("37a1")

magma: E := EllipticCurve([0, 0, 1, -1, 0]); // or

magma: E := EllipticCurve("37a1");

$$y^2 + y = x^{3} - x$$

## Mordell-Weil group structure

$$\Z$$

### Infinite order Mordell-Weil generator and height

sage: E.gens()

magma: Generators(E);

 $$P$$ = $$\left(0, 0\right)$$ $$\hat{h}(P)$$ ≈ $0.05111140823996884$

## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$\left(-1, 0\right)$$, $$\left(-1, -1\right)$$, $$\left(0, 0\right)$$, $$\left(0, -1\right)$$, $$\left(1, 0\right)$$, $$\left(1, -1\right)$$, $$\left(2, 2\right)$$, $$\left(2, -3\right)$$, $$\left(6, 14\right)$$, $$\left(6, -15\right)$$

## 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: $\mathrm{SU}(2)$

## BSD invariants

 sage: E.rank()  magma: Rank(E); Rank: $$1$$ sage: E.regulator()  magma: Regulator(E); Regulator: $$0.05111140824$$ sage: E.period_lattice().omega()  gp: E.omega[1]  magma: RealPeriod(E); Real period: $$5.98691729246$$ 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: $$1$$  = $$1$$ 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)

## Modular invariants

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);

$$q - 2q^{2} - 3q^{3} + 2q^{4} - 2q^{5} + 6q^{6} - q^{7} + 6q^{9} + 4q^{10} - 5q^{11} - 6q^{12} - 2q^{13} + 2q^{14} + 6q^{15} - 4q^{16} - 12q^{18} + O(q^{20})$$

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 2 $$\Gamma_0(N)$$-optimal: yes Manin constant: 1

#### 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);

$$L'(E,1)$$ ≈ $$0.305999773834$$

## Local data

This elliptic curve is semistable.

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

The 2-adic representation attached to this elliptic curve is surjective.

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 $$p$$ .

## $p$-adic data

### $p$-adic regulators

sage: [E.padic_regulator(p) for p in primes(3,20) if E.conductor().valuation(p)<2]

Note: $$p$$-adic regulator data only exists for primes $$p\ge5$$ of good ordinary reduction.

## Iwasawa invariants

 $p$ Reduction type $\lambda$-invariant(s) $\mu$-invariant(s) 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 ss ss ordinary ordinary ordinary ordinary ss ss ordinary ordinary ordinary nonsplit ordinary ordinary ordinary 2,1 1,5 1 1 1 3 1,3 1,1 1 1 1 1 1 1 1 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 37a consists of this curve only.

## Growth of torsion in number fields

The number fields $K$ of degree up to 7 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 $$\Z/2\Z \times \Z/2\Z$$ Not in database

We only show fields where the torsion growth is primitive. For each field $K$ we either show its label, or a defining polynomial when $K$ is not in the database.

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].