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Properties

Label 678.d
Number of curves $2$
Conductor $678$
CM no
Rank $0$
Graph

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Show commands: SageMath
E = EllipticCurve("d1")
 
E.isogeny_class()
 

Elliptic curves in class 678.d

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
678.d1 678f1 \([1, 0, 0, -190, -1024]\) \(758800078561/36612\) \(36612\) \([2]\) \(288\) \(-0.051439\) \(\Gamma_0(N)\)-optimal
678.d2 678f2 \([1, 0, 0, -180, -1134]\) \(-645196518721/167554818\) \(-167554818\) \([2]\) \(576\) \(0.29513\)  

Rank

sage: E.rank()
 

The elliptic curves in class 678.d have rank \(0\).

Complex multiplication

The elliptic curves in class 678.d do not have complex multiplication.

Modular form 678.2.a.d

sage: E.q_eigenform(10)
 
\(q + q^{2} + q^{3} + q^{4} - 4 q^{5} + q^{6} + 4 q^{7} + q^{8} + q^{9} - 4 q^{10} + 4 q^{11} + q^{12} - 2 q^{13} + 4 q^{14} - 4 q^{15} + q^{16} + 6 q^{17} + q^{18} + O(q^{20})\) 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 LMFDB 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 LMFDB labels.