# Properties

 Label 97461.w Number of curves $2$ Conductor $97461$ CM no Rank $1$ Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("w1")

sage: E.isogeny_class()

## Elliptic curves in class 97461.w

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
97461.w1 97461w2 $$[1, -1, 0, -5167500, 4522649543]$$ $$177930109857804849/634933$$ $$54455740504893$$ $$$$ $$2764800$$ $$2.2774$$
97461.w2 97461w1 $$[1, -1, 0, -323115, 70659728]$$ $$43499078731809/82055753$$ $$7037603640544113$$ $$$$ $$1382400$$ $$1.9309$$ $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

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

## Complex multiplication

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

## Modular form 97461.2.a.w

sage: E.q_eigenform(10)

$$q + q^{2} - q^{4} + 4q^{5} - 3q^{8} + 4q^{10} - 6q^{11} + q^{13} - q^{16} + q^{17} - 8q^{19} + O(q^{20})$$ ## 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. 