# Properties

 Label 97682.h Number of curves 4 Conductor 97682 CM no Rank 1 Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("97682.h1")

sage: E.isogeny_class()

## Elliptic curves in class 97682.h

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
97682.h1 97682e4 [1, 1, 0, -5520050, -3451825606]  7962624
97682.h2 97682e3 [1, 1, 0, -5031640, -4345713588]  3981312
97682.h3 97682e2 [1, 1, 0, -2101180, 1171170408]  2654208
97682.h4 97682e1 [1, 1, 0, -147540, 13443344]  1327104 $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

The elliptic curves in class 97682.h have rank $$1$$.

## Modular form 97682.2.a.h

sage: E.q_eigenform(10)

$$q - q^{2} + 2q^{3} + q^{4} - 2q^{6} - 4q^{7} - q^{8} + q^{9} + 6q^{11} + 2q^{12} + 4q^{14} + q^{16} - q^{18} + 4q^{19} + O(q^{20})$$

## 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.

$$\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 