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Properties

Label 488400cb
Number of curves $4$
Conductor $488400$
CM no
Rank $2$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("cb1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 488400cb have rank \(2\).

Complex multiplication

The elliptic curves in class 488400cb do not have complex multiplication.

Modular form 488400.2.a.cb

Copy content sage:E.q_eigenform(10)
 
\(q - q^{3} + q^{9} + q^{11} - 2 q^{13} - 2 q^{17} + 4 q^{19} + O(q^{20})\) 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 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 488400cb

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
488400.cb3 488400cb1 \([0, -1, 0, -10183, -392138]\) \(467147020288/1221\) \(305250000\) \([2]\) \(458752\) \(0.86410\) \(\Gamma_0(N)\)-optimal*
488400.cb2 488400cb2 \([0, -1, 0, -10308, -381888]\) \(30285104848/1490841\) \(5963364000000\) \([2, 2]\) \(917504\) \(1.2107\) \(\Gamma_0(N)\)-optimal*
488400.cb1 488400cb3 \([0, -1, 0, -28808, 1394112]\) \(165256339972/43879077\) \(702065232000000\) \([2]\) \(1835008\) \(1.5572\) \(\Gamma_0(N)\)-optimal*
488400.cb4 488400cb4 \([0, -1, 0, 6192, -1503888]\) \(1640689628/61847313\) \(-989557008000000\) \([2]\) \(1835008\) \(1.5572\)  
*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 488400cb1.