Properties

Label 240c
Number of curves $4$
Conductor $240$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("c1")
 
E.isogeny_class()
 

Elliptic curves in class 240c

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
240.a4 240c1 \([0, -1, 0, 4, 0]\) \(21296/15\) \(-3840\) \([2]\) \(16\) \(-0.62374\) \(\Gamma_0(N)\)-optimal
240.a3 240c2 \([0, -1, 0, -16, 16]\) \(470596/225\) \(230400\) \([2, 2]\) \(32\) \(-0.27717\)  
240.a2 240c3 \([0, -1, 0, -136, -560]\) \(136835858/1875\) \(3840000\) \([2]\) \(64\) \(0.069403\)  
240.a1 240c4 \([0, -1, 0, -216, 1296]\) \(546718898/405\) \(829440\) \([2]\) \(64\) \(0.069403\)  

Rank

sage: E.rank()
 

The elliptic curves in class 240c have rank \(1\).

Complex multiplication

The elliptic curves in class 240c do not have complex multiplication.

Modular form 240.2.a.c

sage: E.q_eigenform(10)
 
\(q - q^{3} - q^{5} - 4 q^{7} + q^{9} - 6 q^{13} + q^{15} - 2 q^{17} - 4 q^{19} + O(q^{20})\) 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.

\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.