Properties

Label 68445bf
Number of curves $2$
Conductor $68445$
CM no
Rank $0$
Graph

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Show commands: SageMath
E = EllipticCurve("bf1")
 
E.isogeny_class()
 

Elliptic curves in class 68445bf

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
68445.x2 68445bf1 \([1, -1, 0, -285, -57394]\) \(-9/5\) \(-1425091223205\) \([]\) \(82944\) \(1.0112\) \(\Gamma_0(N)\)-optimal
68445.x1 68445bf2 \([1, -1, 0, -342510, 77572925]\) \(-15590912409/78125\) \(-22267050362578125\) \([]\) \(580608\) \(1.9841\)  

Rank

sage: E.rank()
 

The elliptic curves in class 68445bf have rank \(0\).

Complex multiplication

The elliptic curves in class 68445bf do not have complex multiplication.

Modular form 68445.2.a.bf

sage: E.q_eigenform(10)
 
\(q + q^{2} - q^{4} - q^{5} + 3 q^{7} - 3 q^{8} - q^{10} - 2 q^{11} + 3 q^{14} - q^{16} - 4 q^{17} + 8 q^{19} + O(q^{20})\) 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.

\(\left(\begin{array}{rr} 1 & 7 \\ 7 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.