# Properties

 Label 88725s Number of curves 4 Conductor 88725 CM no Rank 0 Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("88725.cb1")

sage: E.isogeny_class()

## Elliptic curves in class 88725s

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
88725.cb3 88725s1 [1, 1, 0, -10650, 396375]  184320 $$\Gamma_0(N)$$-optimal
88725.cb2 88725s2 [1, 1, 0, -31775, -1695000] [2, 2] 368640
88725.cb4 88725s3 [1, 1, 0, 73850, -10461875]  737280
88725.cb1 88725s4 [1, 1, 0, -475400, -126353625]  737280

## Rank

sage: E.rank()

The elliptic curves in class 88725s have rank $$0$$.

## Modular form 88725.2.a.cb

sage: E.q_eigenform(10)

$$q + q^{2} - q^{3} - q^{4} - q^{6} + q^{7} - 3q^{8} + q^{9} + q^{12} + q^{14} - q^{16} - 2q^{17} + q^{18} + 8q^{19} + O(q^{20})$$

## 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 Cremona numbering.

$$\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with Cremona labels. 