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SageMath
E = EllipticCurve("a1")
E.isogeny_class()
Elliptic curves in class 363a
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 363.b3 | 363a1 | \([1, 1, 1, -789, 8130]\) | \(30664297/297\) | \(526153617\) | \([4]\) | \(180\) | \(0.49317\) | \(\Gamma_0(N)\)-optimal |
| 363.b2 | 363a2 | \([1, 1, 1, -1394, -6874]\) | \(169112377/88209\) | \(156267624249\) | \([2, 2]\) | \(360\) | \(0.83974\) | |
| 363.b1 | 363a3 | \([1, 1, 1, -17729, -915100]\) | \(347873904937/395307\) | \(700310464227\) | \([2]\) | \(720\) | \(1.1863\) | |
| 363.b4 | 363a4 | \([1, 1, 1, 5261, -46804]\) | \(9090072503/5845851\) | \(-10356281643411\) | \([2]\) | \(720\) | \(1.1863\) |
Rank
sage: E.rank()
The elliptic curves in class 363a have rank \(0\).
Complex multiplication
The elliptic curves in class 363a do not have complex multiplication.Modular form 363.2.a.a
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.
