Properties

 Label 405042.y Number of curves $2$ Conductor $405042$ CM no Rank $1$ Graph

Related objects

Show commands for: SageMath
sage: E = EllipticCurve("y1")

sage: E.isogeny_class()

Elliptic curves in class 405042.y

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
405042.y1 405042y2 [1, 1, 0, -65709, -6510285] [2] 1368576
405042.y2 405042y1 [1, 1, 0, -4339, -90983] [2] 684288 $$\Gamma_0(N)$$-optimal*
*optimality has not been determined rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 0 curves highlighted, and conditionally curve 405042.y1.

Rank

sage: E.rank()

The elliptic curves in class 405042.y have rank $$1$$.

Complex multiplication

The elliptic curves in class 405042.y do not have complex multiplication.

Modular form 405042.2.a.y

sage: E.q_eigenform(10)

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

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 LMFDB 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 LMFDB labels.