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SageMath
E = EllipticCurve("b1")
E.isogeny_class()
Elliptic curves in class 144b
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 144.b5 | 144b1 | \([0, 0, 0, 6, 7]\) | \(2048/3\) | \(-34992\) | \([2]\) | \(8\) | \(-0.44262\) | \(\Gamma_0(N)\)-optimal |
| 144.b4 | 144b2 | \([0, 0, 0, -39, 70]\) | \(35152/9\) | \(1679616\) | \([2, 2]\) | \(16\) | \(-0.096046\) | |
| 144.b3 | 144b3 | \([0, 0, 0, -219, -1190]\) | \(1556068/81\) | \(60466176\) | \([2, 2]\) | \(32\) | \(0.25053\) | |
| 144.b2 | 144b4 | \([0, 0, 0, -579, 5362]\) | \(28756228/3\) | \(2239488\) | \([4]\) | \(32\) | \(0.25053\) | |
| 144.b1 | 144b5 | \([0, 0, 0, -3459, -78302]\) | \(3065617154/9\) | \(13436928\) | \([2]\) | \(64\) | \(0.59710\) | |
| 144.b6 | 144b6 | \([0, 0, 0, 141, -4718]\) | \(207646/6561\) | \(-9795520512\) | \([2]\) | \(64\) | \(0.59710\) |
Rank
sage: E.rank()
The elliptic curves in class 144b have rank \(0\).
Complex multiplication
The elliptic curves in class 144b do not have complex multiplication.Modular form 144.2.a.b
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 2 & 2 \\ 4 & 2 & 4 & 1 & 8 & 8 \\ 8 & 4 & 2 & 8 & 1 & 4 \\ 8 & 4 & 2 & 8 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
