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Properties

Label 40425df
Number of curves $2$
Conductor $40425$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("df1")
 
E.isogeny_class()
 

Elliptic curves in class 40425df

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
40425.z1 40425df1 \([1, 0, 0, -74138, 6498267]\) \(571787/99\) \(7802748228515625\) \([2]\) \(250880\) \(1.7696\) \(\Gamma_0(N)\)-optimal
40425.z2 40425df2 \([1, 0, 0, 140237, 37153892]\) \(3869893/9801\) \(-772472074623046875\) \([2]\) \(501760\) \(2.1162\)  

Rank

sage: E.rank()
 

The elliptic curves in class 40425df have rank \(1\).

Complex multiplication

The elliptic curves in class 40425df do not have complex multiplication.

Modular form 40425.2.a.df

sage: E.q_eigenform(10)
 
\(q - q^{2} + q^{3} - q^{4} - q^{6} + 3 q^{8} + q^{9} + q^{11} - q^{12} - 6 q^{13} - q^{16} + 2 q^{17} - q^{18} - 4 q^{19} + O(q^{20})\) Copy content 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 Cremona 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 Cremona labels.