Properties

Label 57498i
Number of curves $2$
Conductor $57498$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("i1")
 
E.isogeny_class()
 

Elliptic curves in class 57498i

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
57498.j2 57498i1 \([1, 0, 1, -65741, -7096840]\) \(-12246522625/1379952\) \(-3540579289552368\) \([2]\) \(437760\) \(1.7206\) \(\Gamma_0(N)\)-optimal
57498.j1 57498i2 \([1, 0, 1, -1078801, -431366368]\) \(54117385890625/587412\) \(1507138481363508\) \([2]\) \(875520\) \(2.0672\)  

Rank

sage: E.rank()
 

The elliptic curves in class 57498i have rank \(1\).

Complex multiplication

The elliptic curves in class 57498i do not have complex multiplication.

Modular form 57498.2.a.i

sage: E.q_eigenform(10)
 
\(q - q^{2} + q^{3} + q^{4} - q^{6} + q^{7} - q^{8} + q^{9} + 4 q^{11} + q^{12} + 4 q^{13} - q^{14} + q^{16} - q^{18} - 6 q^{19} + O(q^{20})\) 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.