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SageMath
E = EllipticCurve("ew1")
E.isogeny_class()
Elliptic curves in class 121968ew
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
121968.h1 | 121968ew1 | \([0, 0, 0, -91839, 10922186]\) | \(-2141392/49\) | \(-1960221078579456\) | \([]\) | \(760320\) | \(1.7220\) | \(\Gamma_0(N)\)-optimal |
121968.h2 | 121968ew2 | \([0, 0, 0, 387321, 47817506]\) | \(160630448/117649\) | \(-4706490809669273856\) | \([]\) | \(2280960\) | \(2.2713\) |
Rank
sage: E.rank()
The elliptic curves in class 121968ew have rank \(2\).
Complex multiplication
The elliptic curves in class 121968ew do not have complex multiplication.Modular form 121968.2.a.ew
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.
\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.