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Properties

Label 418275.g
Number of curves $2$
Conductor $418275$
CM no
Rank $1$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("g1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 418275.g have rank \(1\).

Complex multiplication

The elliptic curves in class 418275.g do not have complex multiplication.

Modular form 418275.2.a.g

Copy content sage:E.q_eigenform(10)
 
\(q - q^{2} - q^{4} - 5 q^{7} + 3 q^{8} + q^{11} + 5 q^{14} - q^{16} + 6 q^{19} + O(q^{20})\) 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.

\(\left(\begin{array}{rr} 1 & 7 \\ 7 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.

Elliptic curves in class 418275.g

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
418275.g1 418275g2 \([1, -1, 1, -43312730, 109722874772]\) \(4668056654282578921/213092214885\) \(410205843219482578125\) \([]\) \(37933056\) \(3.0311\) \(\Gamma_0(N)\)-optimal*
418275.g2 418275g1 \([1, -1, 1, -827105, -289300228]\) \(32506551525721/2578125\) \(4962930908203125\) \([]\) \(5419008\) \(2.0582\) \(\Gamma_0(N)\)-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 418275.g1.