Properties

Label 53312bd
Number of curves $2$
Conductor $53312$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("bd1")
 
E.isogeny_class()
 

Elliptic curves in class 53312bd

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
53312.g2 53312bd1 \([0, 1, 0, -188225, -3750881]\) \(23912763841/13647872\) \(420913739970117632\) \([2]\) \(1032192\) \(2.0715\) \(\Gamma_0(N)\)-optimal
53312.g1 53312bd2 \([0, 1, 0, -2195265, -1250122721]\) \(37936442980801/88817792\) \(2739227698399281152\) \([2]\) \(2064384\) \(2.4181\)  

Rank

sage: E.rank()
 

The elliptic curves in class 53312bd have rank \(1\).

Complex multiplication

The elliptic curves in class 53312bd do not have complex multiplication.

Modular form 53312.2.a.bd

sage: E.q_eigenform(10)
 
\(q - 2 q^{3} - 4 q^{5} + q^{9} + 6 q^{11} - 2 q^{13} + 8 q^{15} + q^{17} + O(q^{20})\) 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.