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SageMath
E = EllipticCurve("a1")
E.isogeny_class()
Elliptic curves in class 48.a
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 48.a1 | 48a5 | \([0, 1, 0, -384, 2772]\) | \(3065617154/9\) | \(18432\) | \([4]\) | \(8\) | \(0.047795\) | |
| 48.a2 | 48a2 | \([0, 1, 0, -64, -220]\) | \(28756228/3\) | \(3072\) | \([2]\) | \(4\) | \(-0.29878\) | |
| 48.a3 | 48a3 | \([0, 1, 0, -24, 36]\) | \(1556068/81\) | \(82944\) | \([2, 4]\) | \(4\) | \(-0.29878\) | |
| 48.a4 | 48a1 | \([0, 1, 0, -4, -4]\) | \(35152/9\) | \(2304\) | \([2, 2]\) | \(2\) | \(-0.64535\) | \(\Gamma_0(N)\)-optimal |
| 48.a5 | 48a4 | \([0, 1, 0, 1, 0]\) | \(2048/3\) | \(-48\) | \([2]\) | \(4\) | \(-0.99193\) | |
| 48.a6 | 48a6 | \([0, 1, 0, 16, 180]\) | \(207646/6561\) | \(-13436928\) | \([8]\) | \(8\) | \(0.047795\) |
Rank
sage: E.rank()
The elliptic curves in class 48.a have rank \(0\).
Complex multiplication
The elliptic curves in class 48.a do not have complex multiplication.Modular form 48.2.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 LMFDB numbering.
\(\left(\begin{array}{rrrrrr} 1 & 8 & 2 & 4 & 8 & 4 \\ 8 & 1 & 4 & 2 & 4 & 8 \\ 2 & 4 & 1 & 2 & 4 & 2 \\ 4 & 2 & 2 & 1 & 2 & 4 \\ 8 & 4 & 4 & 2 & 1 & 8 \\ 4 & 8 & 2 & 4 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
