# Properties

 Label 61347j Number of curves 4 Conductor 61347 CM no Rank 0 Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("61347.w1")

sage: E.isogeny_class()

## Elliptic curves in class 61347j

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
61347.w3 61347j1 [1, 1, 0, -133344, 18528723]  414720 $$\Gamma_0(N)$$-optimal
61347.w2 61347j2 [1, 1, 0, -235589, -13923840] [2, 2] 829440
61347.w4 61347j3 [1, 1, 0, 889106, -107273525]  1658880
61347.w1 61347j4 [1, 1, 0, -2996204, -1995493287]  1658880

## Rank

sage: E.rank()

The elliptic curves in class 61347j have rank $$0$$.

## Modular form 61347.2.a.w

sage: E.q_eigenform(10)

$$q + q^{2} - q^{3} - q^{4} + 2q^{5} - q^{6} + 4q^{7} - 3q^{8} + q^{9} + 2q^{10} + q^{12} + 4q^{14} - 2q^{15} - q^{16} + 2q^{17} + q^{18} + O(q^{20})$$

## 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. 