Properties

Label 135240bd
Number of curves $2$
Conductor $135240$
CM no
Rank $0$
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Show commands: SageMath
E = EllipticCurve("bd1")
 
E.isogeny_class()
 

Elliptic curves in class 135240bd

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
135240.z2 135240bd1 \([0, -1, 0, -9649880, -11534792100]\) \(824899990643380516/312440625\) \(37640526940800000\) \([2]\) \(3317760\) \(2.5310\) \(\Gamma_0(N)\)-optimal
135240.z1 135240bd2 \([0, -1, 0, -9694960, -11421533108]\) \(418257395996078018/8023271484375\) \(1933168367340000000000\) \([2]\) \(6635520\) \(2.8775\)  

Rank

sage: E.rank()
 

The elliptic curves in class 135240bd have rank \(0\).

Complex multiplication

The elliptic curves in class 135240bd do not have complex multiplication.

Modular form 135240.2.a.bd

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