# Properties

 Label 52371.f Number of curves 3 Conductor 52371 CM no Rank 0 Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("52371.f1")

sage: E.isogeny_class()

## Elliptic curves in class 52371.f

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
52371.f1 52371f3 [0, 0, 1, -37232607, -87444657887] [] 1897500
52371.f2 52371f2 [0, 0, 1, -49197, -7607417] [] 379500
52371.f3 52371f1 [0, 0, 1, -1587, 57793] [] 75900 $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

The elliptic curves in class 52371.f have rank $$0$$.

## Modular form 52371.2.a.f

sage: E.q_eigenform(10)

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

## 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 LMFDB numbering.

$$\left(\begin{array}{rrr} 1 & 5 & 25 \\ 5 & 1 & 5 \\ 25 & 5 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 