# Properties

 Label 83259l Number of curves 4 Conductor 83259 CM no Rank 1 Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("83259.m1")

sage: E.isogeny_class()

## Elliptic curves in class 83259l

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
83259.m3 83259l1 [1, -1, 0, -49356, 4197339]  290304 $$\Gamma_0(N)$$-optimal
83259.m2 83259l2 [1, -1, 0, -87201, -3091608] [2, 2] 580608
83259.m4 83259l3 [1, -1, 0, 329094, -24322653]  1161216
83259.m1 83259l4 [1, -1, 0, -1109016, -448807311]  1161216

## Rank

sage: E.rank()

The elliptic curves in class 83259l have rank $$1$$.

## Modular form 83259.2.a.m

sage: E.q_eigenform(10)

$$q + q^{2} - q^{4} + 2q^{5} + 4q^{7} - 3q^{8} + 2q^{10} + q^{11} - 2q^{13} + 4q^{14} - q^{16} - 2q^{17} + O(q^{20})$$

## 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 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. 