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SageMath
E = EllipticCurve("f1")
E.isogeny_class()
Elliptic curves in class 122034.f
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
122034.f1 | 122034g1 | \([1, 1, 1, -360835162, -2639154712345]\) | \(-821938895581650775417/282039076306944\) | \(-1782871395340807183712256\) | \([]\) | \(39118464\) | \(3.6247\) | \(\Gamma_0(N)\)-optimal |
122034.f2 | 122034g2 | \([1, 1, 1, 208037423, -9905181771985]\) | \(157520606341736640023/6796261691190411264\) | \(-42961637526025314587362983936\) | \([]\) | \(117355392\) | \(4.1740\) |
Rank
sage: E.rank()
The elliptic curves in class 122034.f have rank \(1\).
Complex multiplication
The elliptic curves in class 122034.f do not have complex multiplication.Modular form 122034.2.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.
\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.