# Properties

 Label 31713e Number of curves 4 Conductor 31713 CM no Rank 1 Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("31713.h1")
sage: E.isogeny_class()

## Elliptic curves in class 31713e

sage: E.isogeny_class().curves
LMFDB label Cremona label Weierstrass coefficients Torsion order Modular degree Optimality
31713.h3 31713e1 [1, 0, 1, -6267, 188809] 2 43200 $$\Gamma_0(N)$$-optimal
31713.h2 31713e2 [1, 0, 1, -11072, -141775] 4 86400
31713.h4 31713e3 [1, 0, 1, 41783, -1093165] 2 172800
31713.h1 31713e4 [1, 0, 1, -140807, -20328541] 2 172800

## Rank

sage: E.rank()

The elliptic curves in class 31713e have rank $$1$$.

## Modular form 31713.2.a.h

sage: E.q_eigenform(10)
$$q + q^{2} + q^{3} - q^{4} - 2q^{5} + q^{6} + 4q^{7} - 3q^{8} + q^{9} - 2q^{10} - q^{11} - q^{12} + 2q^{13} + 4q^{14} - 2q^{15} - q^{16} + 2q^{17} + q^{18} + O(q^{20})$$

## 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. 