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SageMath
E = EllipticCurve("fd1")
E.isogeny_class()
Elliptic curves in class 203280.fd
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
203280.fd1 | 203280bl4 | \([0, 1, 0, -4111136, -3175093836]\) | \(1058993490188089/13182390375\) | \(95655561933313536000\) | \([2]\) | \(8847360\) | \(2.6437\) | |
203280.fd2 | 203280bl2 | \([0, 1, 0, -481136, 49798164]\) | \(1697509118089/833765625\) | \(6050065057344000000\) | \([2, 2]\) | \(4423680\) | \(2.2971\) | |
203280.fd3 | 203280bl1 | \([0, 1, 0, -394016, 94996020]\) | \(932288503609/779625\) | \(5657203689984000\) | \([2]\) | \(2211840\) | \(1.9505\) | \(\Gamma_0(N)\)-optimal |
203280.fd4 | 203280bl3 | \([0, 1, 0, 1754944, 383421300]\) | \(82375335041831/56396484375\) | \(-409230591000000000000\) | \([2]\) | \(8847360\) | \(2.6437\) |
Rank
sage: E.rank()
The elliptic curves in class 203280.fd have rank \(1\).
Complex multiplication
The elliptic curves in class 203280.fd do not have complex multiplication.Modular form 203280.2.a.fd
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.