Properties

Label 216384.d
Number of curves $2$
Conductor $216384$
CM no
Rank $1$
Graph

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

Elliptic curves in class 216384.d

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
216384.d1 216384gw2 \([0, -1, 0, -227425, 10587361]\) \(42180533641/22862322\) \(705096403118456832\) \([2]\) \(3538944\) \(2.1157\)  
216384.d2 216384gw1 \([0, -1, 0, 54815, 1273441]\) \(590589719/365148\) \(-11261521966399488\) \([2]\) \(1769472\) \(1.7691\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 216384.d have rank \(1\).

Complex multiplication

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

Modular form 216384.2.a.d

sage: E.q_eigenform(10)
 
\(q - q^{3} - 4 q^{5} + q^{9} - 2 q^{11} + 2 q^{13} + 4 q^{15} - 2 q^{17} - 2 q^{19} + O(q^{20})\) 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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.