Properties

Label 103488cf
Number of curves $2$
Conductor $103488$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("cf1")
 
E.isogeny_class()
 

Elliptic curves in class 103488cf

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
103488.e2 103488cf1 \([0, -1, 0, -1270145, -860083839]\) \(-7347774183121/6119866368\) \(-188742672928953335808\) \([2]\) \(6193152\) \(2.5884\) \(\Gamma_0(N)\)-optimal
103488.e1 103488cf2 \([0, -1, 0, -23347585, -43403310719]\) \(45637459887836881/13417633152\) \(413812948388976525312\) \([2]\) \(12386304\) \(2.9350\)  

Rank

sage: E.rank()
 

The elliptic curves in class 103488cf have rank \(1\).

Complex multiplication

The elliptic curves in class 103488cf do not have complex multiplication.

Modular form 103488.2.a.cf

sage: E.q_eigenform(10)
 
\(q - q^{3} - 4 q^{5} + q^{9} + q^{11} - 6 q^{13} + 4 q^{15} + 4 q^{17} - 2 q^{19} + O(q^{20})\) 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.