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Properties

Label 462400.ip
Number of curves $2$
Conductor $462400$
CM no
Rank $1$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("ip1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 462400.ip have rank \(1\).

Complex multiplication

The elliptic curves in class 462400.ip do not have complex multiplication.

Modular form 462400.2.a.ip

Copy content sage:E.q_eigenform(10)
 
\(q + 2 q^{3} + 3 q^{7} + q^{9} - 3 q^{11} + 3 q^{13} + 4 q^{19} + O(q^{20})\) 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.

\(\left(\begin{array}{rr} 1 & 17 \\ 17 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.

Elliptic curves in class 462400.ip

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
462400.ip1 462400ip2 \([0, -1, 0, -12216993, -18421278943]\) \(-882216989/131072\) \(-29960650073923649536000\) \([]\) \(39951360\) \(3.0427\)  
462400.ip2 462400ip1 \([0, -1, 0, -194593, 33105057]\) \(-297756989/2\) \(-5473632256000\) \([]\) \(2350080\) \(1.6261\) \(\Gamma_0(N)\)-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 462400.ip1.