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SageMath
E = EllipticCurve("ie1")
E.isogeny_class()
Elliptic curves in class 485100ie
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
485100.ie2 | 485100ie1 | \([0, 0, 0, 1363425, -461592250]\) | \(16674224/15125\) | \(-254253665704500000000\) | \([]\) | \(17418240\) | \(2.6027\) | \(\Gamma_0(N)\)-optimal* |
485100.ie1 | 485100ie2 | \([0, 0, 0, -14071575, 27522062750]\) | \(-18330740176/8857805\) | \(-148901116783183380000000\) | \([]\) | \(52254720\) | \(3.1521\) | \(\Gamma_0(N)\)-optimal* |
Rank
sage: E.rank()
The elliptic curves in class 485100ie have rank \(0\).
Complex multiplication
The elliptic curves in class 485100ie do not have complex multiplication.Modular form 485100.2.a.ie
sage: E.q_eigenform(10)
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.
\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.