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SageMath
E = EllipticCurve("t1")
E.isogeny_class()
Elliptic curves in class 4680t
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 4680.p4 | 4680t1 | \([0, 0, 0, 753, -5614]\) | \(253012016/219375\) | \(-40940640000\) | \([4]\) | \(3072\) | \(0.72382\) | \(\Gamma_0(N)\)-optimal |
| 4680.p3 | 4680t2 | \([0, 0, 0, -3747, -49714]\) | \(7793764996/3080025\) | \(2299226342400\) | \([2, 2]\) | \(6144\) | \(1.0704\) | |
| 4680.p1 | 4680t3 | \([0, 0, 0, -52347, -4608394]\) | \(10625310339698/3855735\) | \(5756581509120\) | \([2]\) | \(12288\) | \(1.4170\) | |
| 4680.p2 | 4680t4 | \([0, 0, 0, -27147, 1686566]\) | \(1481943889298/34543665\) | \(51573415495680\) | \([2]\) | \(12288\) | \(1.4170\) |
Rank
sage: E.rank()
The elliptic curves in class 4680t have rank \(1\).
Complex multiplication
The elliptic curves in class 4680t do not have complex multiplication.Modular form 4680.2.a.t
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
