# Properties

 Label 400752cv Number of curves 2 Conductor 400752 CM no Rank 1 Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("400752.cv1")

sage: E.isogeny_class()

## Elliptic curves in class 400752cv

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
400752.cv2 400752cv1 [0, 0, 0, -401115, 1421909962]  9830400 $$\Gamma_0(N)$$-optimal*
400752.cv1 400752cv2 [0, 0, 0, -21571275, 38262222394]  19660800 $$\Gamma_0(N)$$-optimal*
*optimality has not been proved rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 2 curves highlighted, and conditionally curve 400752cv1.

## Rank

sage: E.rank()

The elliptic curves in class 400752cv have rank $$1$$.

## Modular form 400752.2.a.cv

sage: E.q_eigenform(10)

$$q - 2q^{7} - 2q^{13} + 2q^{19} + O(q^{20})$$

## 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.

$$\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with Cremona labels. 