Properties

Label 216384ez
Number of curves $4$
Conductor $216384$
CM no
Rank $2$
Graph

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Show commands: SageMath
E = EllipticCurve("ez1")
 
E.isogeny_class()
 

Elliptic curves in class 216384ez

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
216384.fj4 216384ez1 \([0, 1, 0, -1878529, 1615933535]\) \(-23771111713777/22848457968\) \(-704668822792367505408\) \([2]\) \(8847360\) \(2.6965\) \(\Gamma_0(N)\)-optimal
216384.fj3 216384ez2 \([0, 1, 0, -35057409, 79858368351]\) \(154502321244119857/55101928644\) \(1699397449295581937664\) \([2, 2]\) \(17694720\) \(3.0431\)  
216384.fj1 216384ez3 \([0, 1, 0, -560870529, 5112415739871]\) \(632678989847546725777/80515134\) \(2483165593591087104\) \([2]\) \(35389440\) \(3.3897\)  
216384.fj2 216384ez4 \([0, 1, 0, -40106369, 55347687135]\) \(231331938231569617/90942310746882\) \(2804749937425755778056192\) \([2]\) \(35389440\) \(3.3897\)  

Rank

sage: E.rank()
 

The elliptic curves in class 216384ez have rank \(2\).

Complex multiplication

The elliptic curves in class 216384ez do not have complex multiplication.

Modular form 216384.2.a.ez

sage: E.q_eigenform(10)
 
\(q + q^{3} - 2 q^{5} + q^{9} - 4 q^{11} + 2 q^{13} - 2 q^{15} - 6 q^{17} + O(q^{20})\) 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.

\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.