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SageMath
E = EllipticCurve("cr1")
E.isogeny_class()
Elliptic curves in class 129960.cr
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 129960.cr1 | 129960bk4 | \([0, 0, 0, -702867, -226662514]\) | \(546718898/405\) | \(28446845206210560\) | \([2]\) | \(1769472\) | \(2.0909\) | |
| 129960.cr2 | 129960bk3 | \([0, 0, 0, -442947, 112117214]\) | \(136835858/1875\) | \(131698357436160000\) | \([2]\) | \(1769472\) | \(2.0909\) | |
| 129960.cr3 | 129960bk2 | \([0, 0, 0, -53067, -1961674]\) | \(470596/225\) | \(7901901446169600\) | \([2, 2]\) | \(884736\) | \(1.7444\) | |
| 129960.cr4 | 129960bk1 | \([0, 0, 0, 11913, -233206]\) | \(21296/15\) | \(-131698357436160\) | \([2]\) | \(442368\) | \(1.3978\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 129960.cr have rank \(0\).
Complex multiplication
The elliptic curves in class 129960.cr do not have complex multiplication.Modular form 129960.2.a.cr
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.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
