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SageMath
E = EllipticCurve("m1")
E.isogeny_class()
Elliptic curves in class 12789.m
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
12789.m1 | 12789q2 | \([0, 0, 1, -948297, -355478769]\) | \(-1099616058781696/143578043\) | \(-12314131808881203\) | \([]\) | \(345600\) | \(2.1078\) | |
12789.m2 | 12789q1 | \([0, 0, 1, 8673, -10719]\) | \(841232384/487403\) | \(-41802664673763\) | \([]\) | \(69120\) | \(1.3031\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 12789.m have rank \(0\).
Complex multiplication
The elliptic curves in class 12789.m do not have complex multiplication.Modular form 12789.2.a.m
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.
\(\left(\begin{array}{rr} 1 & 5 \\ 5 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.