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SageMath
E = EllipticCurve("t1")
E.isogeny_class()
Elliptic curves in class 4680.t
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 4680.t1 | 4680s3 | \([0, 0, 0, -5067, 138726]\) | \(9636491538/8125\) | \(12130560000\) | \([2]\) | \(4096\) | \(0.86243\) | |
| 4680.t2 | 4680s2 | \([0, 0, 0, -387, 1134]\) | \(8586756/4225\) | \(3153945600\) | \([2, 2]\) | \(2048\) | \(0.51585\) | |
| 4680.t3 | 4680s1 | \([0, 0, 0, -207, -1134]\) | \(5256144/65\) | \(12130560\) | \([2]\) | \(1024\) | \(0.16928\) | \(\Gamma_0(N)\)-optimal |
| 4680.t4 | 4680s4 | \([0, 0, 0, 1413, 8694]\) | \(208974222/142805\) | \(-213206722560\) | \([2]\) | \(4096\) | \(0.86243\) |
Rank
sage: E.rank()
The elliptic curves in class 4680.t have rank \(0\).
Complex multiplication
The elliptic curves in class 4680.t 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 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.
