# Properties

 Label 61009.b Number of curves $3$ Conductor $61009$ CM no Rank $2$ Graph # Learn more about

Show commands for: SageMath
sage: E = EllipticCurve("b1")

sage: E.isogeny_class()

## Elliptic curves in class 61009.b

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
61009.b1 61009b3 $$[0, -1, 1, -46936257, 123784336522]$$ $$-50357871050752/19$$ $$-4314548154650851$$ $$[]$$ $$2332800$$ $$2.7881$$
61009.b2 61009b2 $$[0, -1, 1, -569417, 176136937]$$ $$-89915392/6859$$ $$-1557551883828957211$$ $$[]$$ $$777600$$ $$2.2388$$
61009.b3 61009b1 $$[0, -1, 1, 40673, 125972]$$ $$32768/19$$ $$-4314548154650851$$ $$[]$$ $$259200$$ $$1.6895$$ $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

The elliptic curves in class 61009.b have rank $$2$$.

## Complex multiplication

The elliptic curves in class 61009.b do not have complex multiplication.

## Modular form 61009.2.a.b

sage: E.q_eigenform(10)

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