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SageMath
E = EllipticCurve("ce1")
E.isogeny_class()
Elliptic curves in class 25872.ce
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 25872.ce1 | 25872ct4 | \([0, 1, 0, -11080288, -14198981644]\) | \(312196988566716625/25367712678\) | \(12224454774186074112\) | \([2]\) | \(663552\) | \(2.7077\) | |
| 25872.ce2 | 25872ct3 | \([0, 1, 0, -645248, -253594188]\) | \(-61653281712625/21875235228\) | \(-10541463754092429312\) | \([2]\) | \(331776\) | \(2.3611\) | |
| 25872.ce3 | 25872ct2 | \([0, 1, 0, -284608, 29216564]\) | \(5290763640625/2291573592\) | \(1104287094887251968\) | \([2]\) | \(221184\) | \(2.1584\) | |
| 25872.ce4 | 25872ct1 | \([0, 1, 0, 60352, 3413556]\) | \(50447927375/39517632\) | \(-19043163697840128\) | \([2]\) | \(110592\) | \(1.8118\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 25872.ce have rank \(1\).
Complex multiplication
The elliptic curves in class 25872.ce do not have complex multiplication.Modular form 25872.2.a.ce
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 LMFDB numbering.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
