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SageMath
E = EllipticCurve("a1")
E.isogeny_class()
Elliptic curves in class 144.a
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality | CM discriminant |
---|---|---|---|---|---|---|---|---|---|
144.a1 | 144a4 | \([0, 0, 0, -135, 594]\) | \(54000\) | \(5038848\) | \([2]\) | \(24\) | \(0.080464\) | \(-12\) | |
144.a2 | 144a2 | \([0, 0, 0, -15, -22]\) | \(54000\) | \(6912\) | \([2]\) | \(8\) | \(-0.46884\) | \(-12\) | |
144.a3 | 144a1 | \([0, 0, 0, 0, -1]\) | \(0\) | \(-432\) | \([2]\) | \(4\) | \(-0.81542\) | \(\Gamma_0(N)\)-optimal | \(-3\) |
144.a4 | 144a3 | \([0, 0, 0, 0, 27]\) | \(0\) | \(-314928\) | \([2]\) | \(12\) | \(-0.26611\) | \(-3\) |
Rank
sage: E.rank()
The elliptic curves in class 144.a have rank \(0\).
Complex multiplication
Each elliptic curve in class 144.a has complex multiplication by an order in the imaginary quadratic field \(\Q(\sqrt{-3}) \).Modular form 144.2.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.
\(\left(\begin{array}{rrrr} 1 & 3 & 6 & 2 \\ 3 & 1 & 2 & 6 \\ 6 & 2 & 1 & 3 \\ 2 & 6 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.