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SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 33450.e
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
33450.e1 | 33450b2 | \([1, 1, 0, -5656250, -7523411250]\) | \(-1280824409818832580001/822726139895701410\) | \(-12855095935870334531250\) | \([]\) | \(3358656\) | \(2.9436\) | |
33450.e2 | 33450b1 | \([1, 1, 0, -170000, 30000000]\) | \(-34773983355859201/4877010000000\) | \(-76203281250000000\) | \([]\) | \(479808\) | \(1.9707\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 33450.e have rank \(0\).
Complex multiplication
The elliptic curves in class 33450.e do not have complex multiplication.Modular form 33450.2.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 LMFDB numbering.
\(\left(\begin{array}{rr} 1 & 7 \\ 7 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.