Properties

Label 485100.d
Number of curves $2$
Conductor $485100$
CM no
Rank $2$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("d1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 485100.d

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
485100.d1 485100d2 \([0, 0, 0, -5692575, 5218487750]\) \(59466754384/121275\) \(41605145297100000000\) \([2]\) \(17694720\) \(2.6511\) \(\Gamma_0(N)\)-optimal*
485100.d2 485100d1 \([0, 0, 0, -235200, 137671625]\) \(-67108864/343035\) \(-7355195329308750000\) \([2]\) \(8847360\) \(2.3045\) \(\Gamma_0(N)\)-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 485100.d1.

Rank

sage: E.rank()
 

The elliptic curves in class 485100.d have rank \(2\).

Complex multiplication

The elliptic curves in class 485100.d do not have complex multiplication.

Modular form 485100.2.a.d

sage: E.q_eigenform(10)
 
\(q - q^{11} - 6q^{13} - 2q^{17} + O(q^{20})\)  Toggle raw display

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.