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SageMath
E = EllipticCurve("dm1")
E.isogeny_class()
Elliptic curves in class 121968dm
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
121968.y2 | 121968dm1 | \([0, 0, 0, -127776, -144140645]\) | \(-8388608/321489\) | \(-8841944716384869936\) | \([2]\) | \(2027520\) | \(2.3157\) | \(\Gamma_0(N)\)-optimal |
121968.y1 | 121968dm2 | \([0, 0, 0, -4979271, -4253356910]\) | \(31025539568/194481\) | \(85581292069700469504\) | \([2]\) | \(4055040\) | \(2.6623\) |
Rank
sage: E.rank()
The elliptic curves in class 121968dm have rank \(1\).
Complex multiplication
The elliptic curves in class 121968dm do not have complex multiplication.Modular form 121968.2.a.dm
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.