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SageMath
E = EllipticCurve("f1")
E.isogeny_class()
Elliptic curves in class 219351.f
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
219351.f1 | 219351d2 | \([1, 0, 0, -357788, -81762387]\) | \(209849322390625/1882056627\) | \(45428271696119763\) | \([2]\) | \(1658880\) | \(2.0191\) | |
219351.f2 | 219351d1 | \([1, 0, 0, -6653, -3037920]\) | \(-1349232625/164333367\) | \(-3966607984964823\) | \([2]\) | \(829440\) | \(1.6725\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 219351.f have rank \(0\).
Complex multiplication
The elliptic curves in class 219351.f do not have complex multiplication.Modular form 219351.2.a.f
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.