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SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 786e
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
786.d4 | 786e1 | \([1, 1, 0, -29, -3]\) | \(2845178713/1609728\) | \(1609728\) | \([2]\) | \(144\) | \(-0.11906\) | \(\Gamma_0(N)\)-optimal |
786.d2 | 786e2 | \([1, 1, 0, -349, 2365]\) | \(4722184089433/9884736\) | \(9884736\) | \([2, 2]\) | \(288\) | \(0.22752\) | |
786.d1 | 786e3 | \([1, 1, 0, -5589, 158517]\) | \(19312898130234073/84888\) | \(84888\) | \([2]\) | \(576\) | \(0.57409\) | |
786.d3 | 786e4 | \([1, 1, 0, -229, 4165]\) | \(-1337180541913/7067998104\) | \(-7067998104\) | \([2]\) | \(576\) | \(0.57409\) |
Rank
sage: E.rank()
The elliptic curves in class 786e have rank \(0\).
Complex multiplication
The elliptic curves in class 786e do not have complex multiplication.Modular form 786.2.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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.