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SageMath
E = EllipticCurve("n1")
E.isogeny_class()
Elliptic curves in class 3192n
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 3192.d4 | 3192n1 | \([0, -1, 0, -84, -60]\) | \(259108432/136857\) | \(35035392\) | \([4]\) | \(768\) | \(0.13921\) | \(\Gamma_0(N)\)-optimal |
| 3192.d2 | 3192n2 | \([0, -1, 0, -1064, -12996]\) | \(130213720228/159201\) | \(163021824\) | \([2, 2]\) | \(1536\) | \(0.48578\) | |
| 3192.d1 | 3192n3 | \([0, -1, 0, -17024, -849300]\) | \(266442869452034/399\) | \(817152\) | \([2]\) | \(3072\) | \(0.83235\) | |
| 3192.d3 | 3192n4 | \([0, -1, 0, -784, -20276]\) | \(-26055281954/73892007\) | \(-151330830336\) | \([2]\) | \(3072\) | \(0.83235\) |
Rank
sage: E.rank()
The elliptic curves in class 3192n have rank \(1\).
Complex multiplication
The elliptic curves in class 3192n do not have complex multiplication.Modular form 3192.2.a.n
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.
