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Properties

Label 216302.g
Number of curves $3$
Conductor $216302$
CM no
Rank $2$
Graph

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Show commands: SageMath
E = EllipticCurve("g1")
 
E.isogeny_class()
 

Elliptic curves in class 216302.g

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
216302.g1 216302b3 \([1, 0, 0, -7141417, -7346143271]\) \(15698803397448457/20709376\) \(53134592917110784\) \([]\) \(6220800\) \(2.4853\)  
216302.g2 216302b2 \([1, 0, 0, -111602, -4312636]\) \(59914169497/31554496\) \(80960203709884864\) \([]\) \(2073600\) \(1.9360\)  
216302.g3 216302b1 \([1, 0, 0, -63687, 6180749]\) \(11134383337/316\) \(810769545244\) \([]\) \(691200\) \(1.3867\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 216302.g have rank \(2\).

Complex multiplication

The elliptic curves in class 216302.g do not have complex multiplication.

Modular form 216302.2.a.g

sage: E.q_eigenform(10)
 
\(q + q^{2} + q^{3} + q^{4} - 3 q^{5} + q^{6} - q^{7} + q^{8} - 2 q^{9} - 3 q^{10} + q^{12} - 5 q^{13} - q^{14} - 3 q^{15} + q^{16} - 2 q^{18} - 2 q^{19} + O(q^{20})\) 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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.