# Properties

 Label 400710x Number of curves $6$ Conductor $400710$ CM no Rank $0$ Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("400710.x1")

sage: E.isogeny_class()

## Elliptic curves in class 400710x

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
400710.x5 400710x1 [1, 0, 1, -305053, -223267144]  11943936 $$\Gamma_0(N)$$-optimal*
400710.x4 400710x2 [1, 0, 1, -7698333, -8205052232] [2, 2] 23887872 $$\Gamma_0(N)$$-optimal*
400710.x3 400710x3 [1, 0, 1, -10586333, -1489874632] [2, 2] 47775744 $$\Gamma_0(N)$$-optimal*
400710.x1 400710x4 [1, 0, 1, -123102813, -525724902344]  47775744
400710.x2 400710x5 [1, 0, 1, -109428133, 438672429128]  95551488 $$\Gamma_0(N)$$-optimal*
400710.x6 400710x6 [1, 0, 1, 42047467, -11869259992]  95551488
*optimality has not been proved rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 4 curves highlighted, and conditionally curve 400710x1.

## Rank

sage: E.rank()

The elliptic curves in class 400710x have rank $$0$$.

## Modular form 400710.2.a.x

sage: E.q_eigenform(10)

$$q - q^{2} + q^{3} + q^{4} + q^{5} - q^{6} - q^{8} + q^{9} - q^{10} + 4q^{11} + q^{12} + 2q^{13} + q^{15} + q^{16} + 2q^{17} - q^{18} + O(q^{20})$$

## 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.

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

## Isogeny graph

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

The vertices are labelled with Cremona labels. 