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SageMath
E = EllipticCurve("ca1")
E.isogeny_class()
Elliptic curves in class 323400ca
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
323400.ca2 | 323400ca1 | \([0, -1, 0, -160901708, -782934300588]\) | \(7831544736466064/29831377653\) | \(1754815874748898500000000\) | \([2]\) | \(68567040\) | \(3.5108\) | \(\Gamma_0(N)\)-optimal |
323400.ca1 | 323400ca2 | \([0, -1, 0, -2572069208, -50207045715588]\) | \(7997484869919944276/116700507\) | \(27459395896086000000000\) | \([2]\) | \(137134080\) | \(3.8574\) |
Rank
sage: E.rank()
The elliptic curves in class 323400ca have rank \(1\).
Complex multiplication
The elliptic curves in class 323400ca do not have complex multiplication.Modular form 323400.2.a.ca
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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.