Properties

Label 51376.i
Number of curves $3$
Conductor $51376$
CM no
Rank $1$
Graph

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

Elliptic curves in class 51376.i

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
51376.i1 51376v3 \([0, -1, 0, -231248, 344373184]\) \(-69173457625/2550136832\) \(-50417759895261544448\) \([]\) \(886464\) \(2.4608\)  
51376.i2 51376v1 \([0, -1, 0, -41968, -3296320]\) \(-413493625/152\) \(-3005132668928\) \([]\) \(98496\) \(1.3622\) \(\Gamma_0(N)\)-optimal
51376.i3 51376v2 \([0, -1, 0, 25632, -12587264]\) \(94196375/3511808\) \(-69430585182912512\) \([]\) \(295488\) \(1.9115\)  

Rank

sage: E.rank()
 

The elliptic curves in class 51376.i have rank \(1\).

Complex multiplication

The elliptic curves in class 51376.i do not have complex multiplication.

Modular form 51376.2.a.i

sage: E.q_eigenform(10)
 
\(q - q^{3} - q^{7} - 2 q^{9} - 6 q^{11} + 3 q^{17} + q^{19} + O(q^{20})\) 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 LMFDB numbering.

\(\left(\begin{array}{rrr} 1 & 9 & 3 \\ 9 & 1 & 3 \\ 3 & 3 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.