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Properties

Label 447700.g
Number of curves $2$
Conductor $447700$
CM no
Rank $1$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve([0, -1, 0, -36364533, 84416475937]) E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 447700.g have rank \(1\).

Complex multiplication

The elliptic curves in class 447700.g do not have complex multiplication.

Modular form 447700.2.a.g

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

\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.

Elliptic curves in class 447700.g

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
447700.g1 447700g2 \([0, -1, 0, -36364533, 84416475937]\) \(750484394082304/578125\) \(4096734812500000000\) \([]\) \(13996800\) \(2.8793\) \(\Gamma_0(N)\)-optimal*
447700.g2 447700g1 \([0, -1, 0, -548533, 60841937]\) \(2575826944/1266325\) \(8973487933300000000\) \([]\) \(4665600\) \(2.3299\) \(\Gamma_0(N)\)-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 447700.g1.