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SageMath
E = EllipticCurve("ca1")
E.isogeny_class()
Elliptic curves in class 49686.ca
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 49686.ca1 | 49686cg4 | \([1, 1, 1, -16201949, -25103146489]\) | \(828279937799497/193444524\) | \(109851197155307273484\) | \([2]\) | \(3096576\) | \(2.8361\) | |
| 49686.ca2 | 49686cg2 | \([1, 1, 1, -1130529, -295589169]\) | \(281397674377/96589584\) | \(54850254821031340944\) | \([2, 2]\) | \(1548288\) | \(2.4895\) | |
| 49686.ca3 | 49686cg1 | \([1, 1, 1, -468049, 119653295]\) | \(19968681097/628992\) | \(357185216579772672\) | \([4]\) | \(774144\) | \(2.1429\) | \(\Gamma_0(N)\)-optimal |
| 49686.ca4 | 49686cg3 | \([1, 1, 1, 3341211, -2050299945]\) | \(7264187703863/7406095788\) | \(-4205694075675560503308\) | \([2]\) | \(3096576\) | \(2.8361\) |
Rank
sage: E.rank()
The elliptic curves in class 49686.ca have rank \(1\).
Complex multiplication
The elliptic curves in class 49686.ca do not have complex multiplication.Modular form 49686.2.a.ca
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
