# Properties

 Label 418950.pt Number of curves $4$ Conductor $418950$ CM no Rank $1$ Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("418950.pt1")

sage: E.isogeny_class()

## Elliptic curves in class 418950.pt

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
418950.pt1 418950pt3 [1, -1, 1, -33516230, 74692943147]  21233664 $$\Gamma_0(N)$$-optimal*
418950.pt2 418950pt4 [1, -1, 1, -2425730, 774287147]  21233664
418950.pt3 418950pt2 [1, -1, 1, -2094980, 1167218147] [2, 2] 10616832 $$\Gamma_0(N)$$-optimal*
418950.pt4 418950pt1 [1, -1, 1, -110480, 24146147]  5308416 $$\Gamma_0(N)$$-optimal*
*optimality has not been proved rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 3 curves highlighted, and conditionally curve 418950.pt4.

## Rank

sage: E.rank()

The elliptic curves in class 418950.pt have rank $$1$$.

## Modular form 418950.2.a.pt

sage: E.q_eigenform(10)

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

## 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 LMFDB numbering.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 