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SageMath
E = EllipticCurve("ca1")
E.isogeny_class()
Elliptic curves in class 27225ca
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 27225.bz1 | 27225ca1 | \([0, 0, 1, -9075, -341069]\) | \(-102400/3\) | \(-2421502441875\) | \([]\) | \(67200\) | \(1.1552\) | \(\Gamma_0(N)\)-optimal |
| 27225.bz2 | 27225ca2 | \([0, 0, 1, 45375, 16429531]\) | \(20480/243\) | \(-122588561119921875\) | \([]\) | \(336000\) | \(1.9599\) |
Rank
sage: E.rank()
The elliptic curves in class 27225ca have rank \(0\).
Complex multiplication
The elliptic curves in class 27225ca do not have complex multiplication.Modular form 27225.2.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.
\(\left(\begin{array}{rr} 1 & 5 \\ 5 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
