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Properties

Label 485184ij
Number of curves $4$
Conductor $485184$
CM no
Rank $1$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("ij1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 485184ij have rank \(1\).

Complex multiplication

The elliptic curves in class 485184ij do not have complex multiplication.

Modular form 485184.2.a.ij

Copy content sage:E.q_eigenform(10)
 
\(q + q^{3} + 2 q^{5} + q^{7} + q^{9} - 4 q^{11} + 2 q^{13} + 2 q^{15} - 6 q^{17} + O(q^{20})\) 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 Cremona numbering.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.

Elliptic curves in class 485184ij

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
485184.ij4 485184ij1 \([0, 1, 0, -6294877, -2383163053]\) \(572616640141312/280535480757\) \(13514791776226930603008\) \([2]\) \(35389440\) \(2.9398\) \(\Gamma_0(N)\)-optimal*
485184.ij2 485184ij2 \([0, 1, 0, -53665297, 149647462895]\) \(22174957026242512/278654127129\) \(214786521580663695753216\) \([2, 2]\) \(70778880\) \(3.2864\) \(\Gamma_0(N)\)-optimal*
485184.ij1 485184ij3 \([0, 1, 0, -856038337, 9639955305407]\) \(22501000029889239268/3620708343\) \(11163363825450036953088\) \([2]\) \(141557760\) \(3.6329\) \(\Gamma_0(N)\)-optimal*
485184.ij3 485184ij4 \([0, 1, 0, -9218977, 390057607775]\) \(-28104147578308/21301741002339\) \(-65677503515122815183028224\) \([2]\) \(141557760\) \(3.6329\)  
*optimality has not been determined rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 3 curves highlighted, and conditionally curve 485184ij1.