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Properties

Label 471900cd
Number of curves $2$
Conductor $471900$
CM no
Rank $1$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("cd1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 471900cd have rank \(1\).

Complex multiplication

The elliptic curves in class 471900cd do not have complex multiplication.

Modular form 471900.2.a.cd

Copy content sage:E.q_eigenform(10)
 
\(q - q^{3} + 2 q^{7} + q^{9} + q^{13} + 2 q^{17} + 4 q^{19} + O(q^{20})\) 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.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.

Elliptic curves in class 471900cd

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
471900.cd2 471900cd1 \([0, -1, 0, -575153, -77828298]\) \(5938660917248/2694544281\) \(9547099121985282000\) \([2]\) \(8847360\) \(2.3370\) \(\Gamma_0(N)\)-optimal*
471900.cd1 471900cd2 \([0, -1, 0, -4601428, 3747132952]\) \(190062137800208/3079189971\) \(174559131654871392000\) \([2]\) \(17694720\) \(2.6836\) \(\Gamma_0(N)\)-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 471900cd1.