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
Minimal Weierstrass equation
Simplified equation
\(y^2+y=x^3-x\) | (homogenize, simplify) |
\(y^2z+yz^2=x^3-xz^2\) | (dehomogenize, simplify) |
\(y^2=x^3-16x+16\) | (homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Infinite order Mordell-Weil generator and height
$P$ | = | \(\left(0, 0\right)\) |
$\hat{h}(P)$ | ≈ | $0.051111408239968840235886099757$ |
Integral points
\( \left(-1, 0\right) \), \( \left(-1, -1\right) \), \( \left(0, 0\right) \), \( \left(0, -1\right) \), \( \left(1, 0\right) \), \( \left(1, -1\right) \), \( \left(2, 2\right) \), \( \left(2, -3\right) \), \( \left(6, 14\right) \), \( \left(6, -15\right) \)
Invariants
Conductor: | \( 37 \) | = | $37$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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Discriminant: | $37 $ | = | $37 $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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j-invariant: | \( \frac{110592}{37} \) | = | $2^{12} \cdot 3^{3} \cdot 37^{-1}$ | comment: j-invariant
sage: E.j_invariant().factor()
gp: E.j
magma: jInvariant(E);
oscar: j_invariant(E)
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Endomorphism ring: | $\Z$ | |||
Geometric endomorphism ring: | \(\Z\) | (no potential complex multiplication) | sage: E.has_cm()
magma: HasComplexMultiplication(E);
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Sato-Tate group: | $\mathrm{SU}(2)$ | |||
Faltings height: | $-0.99654220763736714794656344357\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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Stable Faltings height: | $-0.99654220763736714794656344357\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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BSD invariants
Analytic rank: | $1$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
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Regulator: | $0.051111408239968840235886099757\dots$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
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Real period: | $5.9869172924639192596640199589\dots$ | comment: Real Period
sage: E.period_lattice().omega()
gp: if(E.disc>0,2,1)*E.omega[1]
magma: (Discriminant(E) gt 0 select 2 else 1) * RealPeriod(E);
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Tamagawa product: | $ 1 $ | comment: Tamagawa numbers
sage: E.tamagawa_numbers()
gp: gr=ellglobalred(E); [[gr[4][i,1],gr[5][i][4]] | i<-[1..#gr[4][,1]]]
magma: TamagawaNumbers(E);
oscar: tamagawa_numbers(E)
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Torsion order: | $1$ | comment: Torsion order
sage: E.torsion_order()
gp: elltors(E)[1]
magma: Order(TorsionSubgroup(E));
oscar: prod(torsion_structure(E)[1])
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Analytic order of Ш: | $1$ ( rounded) | comment: Order of Sha
sage: E.sha().an_numerical()
magma: MordellWeilShaInformation(E);
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Special value: | $ L'(E,1) $ ≈ $ 0.30599977383405230182048368332 $ | comment: Special L-value
r = E.rank();
gp: [r,L1r] = ellanalyticrank(E); L1r/r!
magma: Lr1 where r,Lr1 := AnalyticRank(E: Precision:=12);
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BSD formula
$\displaystyle 0.305999774 \approx L'(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 5.986917 \cdot 0.051111 \cdot 1}{1^2} \approx 0.305999774$
Modular invariants
For more coefficients, see the Downloads section to the right.
Modular degree: | 2 | comment: Modular degree
sage: E.modular_degree()
gp: ellmoddegree(E)
magma: ModularDegree(E);
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$ \Gamma_0(N) $-optimal: | yes | |
Manin constant: | 1 | comment: Manin constant
magma: ManinConstant(E);
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Local data
This elliptic curve is semistable. There is only one prime of bad reduction:
prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
---|---|---|---|---|---|---|---|
$37$ | $1$ | $I_{1}$ | Non-split multiplicative | 1 | 1 | 1 | 1 |
Galois representations
The $\ell$-adic Galois representation has maximal image for all primes $\ell$.
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 74 = 2 \cdot 37 \), index $2$, genus $0$, and generators
$\left(\begin{array}{rr} 39 & 2 \\ 39 & 3 \end{array}\right),\left(\begin{array}{rr} 73 & 2 \\ 72 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 2 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 1 \\ 73 & 0 \end{array}\right)$.
The torsion field $K:=\Q(E[74])$ is a degree-$5466528$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/74\Z)$.
Isogenies
This curve has no rational isogenies. Its isogeny class 37a consists of this curve only.
Twists
This elliptic curve is its own minimal quadratic twist.
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.
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 |
$p$-adic regulators
Note: $p$-adic regulator data only exists for primes $p\ge 5$ of good ordinary reduction.
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 $nP$ 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 $x^3-x = (x-1)x(x+1)$]. That $210 = 5 \cdot 6 \cdot 7 = 14 \cdot 15$ is the last such example follows from the fact that $(0,0)$ generates the group of rational solutions; see page 275, exercise 9.13 of The Arithmetic of Elliptic Curves [10.1007/978-0-387-09494-6].