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SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 354.e
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
354.e1 | 354a1 | \([1, 1, 1, -3, -3]\) | \(3048625/708\) | \(708\) | \([2]\) | \(16\) | \(-0.74108\) | \(\Gamma_0(N)\)-optimal |
354.e2 | 354a2 | \([1, 1, 1, 7, -7]\) | \(37595375/62658\) | \(-62658\) | \([2]\) | \(32\) | \(-0.39451\) |
Rank
sage: E.rank()
The elliptic curves in class 354.e have rank \(0\).
Complex multiplication
The elliptic curves in class 354.e do not have complex multiplication.Modular form 354.2.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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.