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SageMath
E = EllipticCurve("cd1")
E.isogeny_class()
Elliptic curves in class 42432.cd
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
42432.cd1 | 42432t4 | \([0, 1, 0, -56577, 5160927]\) | \(305612563186948/663\) | \(43450368\) | \([2]\) | \(98304\) | \(1.1394\) | |
42432.cd2 | 42432t3 | \([0, 1, 0, -4577, 27903]\) | \(161838334948/87947613\) | \(5763734765568\) | \([2]\) | \(98304\) | \(1.1394\) | |
42432.cd3 | 42432t2 | \([0, 1, 0, -3537, 79695]\) | \(298766385232/439569\) | \(7201898496\) | \([2, 2]\) | \(49152\) | \(0.79287\) | |
42432.cd4 | 42432t1 | \([0, 1, 0, -157, 1955]\) | \(-420616192/1456611\) | \(-1491569664\) | \([2]\) | \(24576\) | \(0.44630\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 42432.cd have rank \(1\).
Complex multiplication
The elliptic curves in class 42432.cd do not have complex multiplication.Modular form 42432.2.a.cd
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.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.