# Properties

 Label 18259.a Number of curves $3$ Conductor $18259$ CM no Rank $1$ Graph # Learn more about

Show commands for: SageMath
sage: E = EllipticCurve("a1")

sage: E.isogeny_class()

## Elliptic curves in class 18259.a

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
18259.a1 18259a3 $$[0, -1, 1, -739329, 244930132]$$ $$-50357871050752/19$$ $$-16862569939$$ $$[]$$ $$90720$$ $$1.7504$$
18259.a2 18259a2 $$[0, -1, 1, -8969, 350827]$$ $$-89915392/6859$$ $$-6087387747979$$ $$[]$$ $$30240$$ $$1.2011$$
18259.a3 18259a1 $$[0, -1, 1, 641, 62]$$ $$32768/19$$ $$-16862569939$$ $$[]$$ $$10080$$ $$0.65182$$ $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

The elliptic curves in class 18259.a have rank $$1$$.

## Complex multiplication

The elliptic curves in class 18259.a do not have complex multiplication.

## Modular form 18259.2.a.a

sage: E.q_eigenform(10)

$$q + 2q^{3} - 2q^{4} + 3q^{5} - q^{7} + q^{9} - 3q^{11} - 4q^{12} + 4q^{13} + 6q^{15} + 4q^{16} + 3q^{17} + q^{19} + O(q^{20})$$ ## Isogeny matrix

sage: E.isogeny_class().matrix()

The $$i,j$$ entry is the smallest degree of a cyclic isogeny between the $$i$$-th and $$j$$-th curve in the isogeny class, in the LMFDB numbering.

$$\left(\begin{array}{rrr} 1 & 3 & 9 \\ 3 & 1 & 3 \\ 9 & 3 & 1 \end{array}\right)$$

## Isogeny graph

sage: E.isogeny_graph().plot(edge_labels=True)

The vertices are labelled with LMFDB labels. 