# Properties

 Label 52371d Number of curves 6 Conductor 52371 CM no Rank 0 Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("52371.d1")

sage: E.isogeny_class()

## Elliptic curves in class 52371d

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
52371.d6 52371d1 [1, -1, 0, 147492, -63958869] [2] 540672 $$\Gamma_0(N)$$-optimal
52371.d5 52371d2 [1, -1, 0, -1780713, -824057280] [2, 2] 1081344
52371.d4 52371d3 [1, -1, 0, -6708348, 5801640741] [2] 2162688
52371.d2 52371d4 [1, -1, 0, -27704358, -56119192065] [2, 2] 2162688
52371.d3 52371d5 [1, -1, 0, -26918793, -59452030134] [2] 4325376
52371.d1 52371d6 [1, -1, 0, -443268243, -3591986063976] [2] 4325376

## Rank

sage: E.rank()

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

## Modular form 52371.2.a.d

sage: E.q_eigenform(10)

$$q + q^{2} - q^{4} - 2q^{5} - 3q^{8} - 2q^{10} + q^{11} - 2q^{13} - q^{16} + 2q^{17} + 4q^{19} + O(q^{20})$$

## 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 Cremona numbering.

$$\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 8 & 8 \\ 4 & 2 & 4 & 1 & 2 & 2 \\ 8 & 4 & 8 & 2 & 1 & 4 \\ 8 & 4 & 8 & 2 & 4 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with Cremona labels.