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SageMath
E = EllipticCurve("df1")
E.isogeny_class()
Elliptic curves in class 485100df
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 485100.df2 | 485100df1 | \([0, 0, 0, -109200, -11553500]\) | \(1007878144/179685\) | \(25674111540000000\) | \([]\) | \(2985984\) | \(1.8680\) | \(\Gamma_0(N)\)-optimal |
| 485100.df1 | 485100df2 | \([0, 0, 0, -8425200, -9412791500]\) | \(462893166690304/4125\) | \(589396500000000\) | \([]\) | \(8957952\) | \(2.4173\) |
Rank
sage: E.rank()
The elliptic curves in class 485100df have rank \(2\).
Complex multiplication
The elliptic curves in class 485100df do not have complex multiplication.Modular form 485100.2.a.df
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.
\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
