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SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 3192e
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
3192.f2 | 3192e1 | \([0, -1, 0, -43, -104]\) | \(-562432000/53067\) | \(-849072\) | \([2]\) | \(384\) | \(-0.12043\) | \(\Gamma_0(N)\)-optimal |
3192.f1 | 3192e2 | \([0, -1, 0, -708, -7020]\) | \(153531250000/1197\) | \(306432\) | \([2]\) | \(768\) | \(0.22614\) |
Rank
sage: E.rank()
The elliptic curves in class 3192e have rank \(1\).
Complex multiplication
The elliptic curves in class 3192e do not have complex multiplication.Modular form 3192.2.a.e
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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.