# Properties

 Label 97461.r Number of curves $2$ Conductor $97461$ CM no Rank $1$ Graph # Learn more about

Show commands for: SageMath
sage: E = EllipticCurve("r1")

sage: E.isogeny_class()

## Elliptic curves in class 97461.r

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
97461.r1 97461a2 $$[1, -1, 0, -242853, -17548210]$$ $$684030715731/338005577$$ $$782714534822734059$$ $$$$ $$1105920$$ $$2.1266$$
97461.r2 97461a1 $$[1, -1, 0, -130398, 17965079]$$ $$105890949891/1288651$$ $$2984110135004817$$ $$$$ $$552960$$ $$1.7800$$ $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

The elliptic curves in class 97461.r have rank $$1$$.

## Complex multiplication

The elliptic curves in class 97461.r do not have complex multiplication.

## Modular form 97461.2.a.r

sage: E.q_eigenform(10)

$$q + q^{2} - q^{4} - 2q^{5} - 3q^{8} - 2q^{10} + q^{13} - q^{16} - q^{17} + 8q^{19} + O(q^{20})$$ ## 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.

$$\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 