Properties

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

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Show commands for: SageMath
sage: E = EllipticCurve("w1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 51376.w

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
51376.w1 51376x3 \([0, -1, 0, -2080277, 1155555101]\) \(-50357871050752/19\) \(-375641583616\) \([]\) \(466560\) \(2.0091\)  
51376.w2 51376x2 \([0, -1, 0, -25237, 1650141]\) \(-89915392/6859\) \(-135606611685376\) \([]\) \(155520\) \(1.4598\)  
51376.w3 51376x1 \([0, -1, 0, 1803, 701]\) \(32768/19\) \(-375641583616\) \([]\) \(51840\) \(0.91045\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

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

Complex multiplication

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

Modular form 51376.2.a.w

sage: E.q_eigenform(10)
 
\(q + 2q^{3} - 3q^{5} - q^{7} + q^{9} + 3q^{11} - 6q^{15} - 3q^{17} + q^{19} + O(q^{20})\)  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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.