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SageMath
E = EllipticCurve("d1")
E.isogeny_class()
Elliptic curves in class 12274d
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 12274.f4 | 12274d1 | \([1, 1, 0, -1090, -9036]\) | \(3048625/1088\) | \(51185918528\) | \([2]\) | \(13824\) | \(0.75583\) | \(\Gamma_0(N)\)-optimal |
| 12274.f3 | 12274d2 | \([1, 1, 0, -15530, -751252]\) | \(8805624625/2312\) | \(108770076872\) | \([2]\) | \(27648\) | \(1.1024\) | |
| 12274.f2 | 12274d3 | \([1, 1, 0, -37190, 2744672]\) | \(120920208625/19652\) | \(924545653412\) | \([2]\) | \(41472\) | \(1.3051\) | |
| 12274.f1 | 12274d4 | \([1, 1, 0, -40800, 2175014]\) | \(159661140625/48275138\) | \(2271146397606578\) | \([2]\) | \(82944\) | \(1.6517\) |
Rank
sage: E.rank()
The elliptic curves in class 12274d have rank \(0\).
Complex multiplication
The elliptic curves in class 12274d do not have complex multiplication.Modular form 12274.2.a.d
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
