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SageMath
E = EllipticCurve("ca1")
E.isogeny_class()
Elliptic curves in class 54978ca
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 54978.bx2 | 54978ca1 | \([1, 0, 0, -20177564, 33530442000]\) | \(7722211175253055152433/340131399900069888\) | \(40016119066843322253312\) | \([2]\) | \(5840640\) | \(3.1000\) | \(\Gamma_0(N)\)-optimal |
| 54978.bx1 | 54978ca2 | \([1, 0, 0, -54297244, -109779037936]\) | \(150476552140919246594353/42832838728685592576\) | \(5039240643591131280973824\) | \([2]\) | \(11681280\) | \(3.4466\) |
Rank
sage: E.rank()
The elliptic curves in class 54978ca have rank \(1\).
Complex multiplication
The elliptic curves in class 54978ca do not have complex multiplication.Modular form 54978.2.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.
