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Properties

Label 83205d
Number of curves $2$
Conductor $83205$
CM no
Rank $0$
Graph

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

Elliptic curves in class 83205d

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
83205.d2 83205d1 \([1, -1, 1, -1300122848, -18041242862294]\) \(1953326569433829507/262451171875\) \(32655064223749273681640625\) \([2]\) \(49674240\) \(3.9146\) \(\Gamma_0(N)\)-optimal
83205.d1 83205d2 \([1, -1, 1, -20801294723, -1154733349581044]\) \(8000051600110940079507/144453125\) \(17973347348751600234375\) \([2]\) \(99348480\) \(4.2611\)  

Rank

sage: E.rank()
 

The elliptic curves in class 83205d have rank \(0\).

Complex multiplication

The elliptic curves in class 83205d do not have complex multiplication.

Modular form 83205.2.a.d

sage: E.q_eigenform(10)
 
\(q - q^{2} - q^{4} - q^{5} - 4 q^{7} + 3 q^{8} + q^{10} + 4 q^{11} + 2 q^{13} + 4 q^{14} - q^{16} + 6 q^{17} + 6 q^{19} + O(q^{20})\) 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 Cremona 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 Cremona labels.