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SageMath
E = EllipticCurve("f1")
E.isogeny_class()
Elliptic curves in class 2670f
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 2670.e3 | 2670f1 | \([1, 0, 0, -1885, 24497]\) | \(740750878754641/166095360000\) | \(166095360000\) | \([4]\) | \(4608\) | \(0.86598\) | \(\Gamma_0(N)\)-optimal |
| 2670.e2 | 2670f2 | \([1, 0, 0, -9885, -357903]\) | \(106820960574626641/6735270657600\) | \(6735270657600\) | \([2, 2]\) | \(9216\) | \(1.2126\) | |
| 2670.e1 | 2670f3 | \([1, 0, 0, -155685, -23656743]\) | \(417315196209220773841/1829563747560\) | \(1829563747560\) | \([2]\) | \(18432\) | \(1.5591\) | |
| 2670.e4 | 2670f4 | \([1, 0, 0, 7915, -1500663]\) | \(54836918279008559/1005449149872360\) | \(-1005449149872360\) | \([2]\) | \(18432\) | \(1.5591\) |
Rank
sage: E.rank()
The elliptic curves in class 2670f have rank \(1\).
Complex multiplication
The elliptic curves in class 2670f do not have complex multiplication.Modular form 2670.2.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 Cremona 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 Cremona labels.
