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Properties

Label 412200.l
Number of curves $2$
Conductor $412200$
CM no
Rank $1$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("l1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 412200.l have rank \(1\).

Complex multiplication

The elliptic curves in class 412200.l do not have complex multiplication.

Modular form 412200.2.a.l

Copy content sage:E.q_eigenform(10)
 
\(q - 2 q^{7} + 4 q^{11} + 6 q^{17} + 4 q^{19} + O(q^{20})\) Copy content Toggle raw display

Isogeny matrix

Copy content 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

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

The vertices are labelled with LMFDB labels.

Elliptic curves in class 412200.l

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
412200.l1 412200l2 \([0, 0, 0, -503175, 136858250]\) \(4831626541264/21238605\) \(61931772180000000\) \([2]\) \(5308416\) \(2.0747\) \(\Gamma_0(N)\)-optimal*
412200.l2 412200l1 \([0, 0, 0, -47550, -284875]\) \(65239066624/37561725\) \(6845624381250000\) \([2]\) \(2654208\) \(1.7281\) \(\Gamma_0(N)\)-optimal*
*optimality has not been determined rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 2 curves highlighted, and conditionally curve 412200.l1.