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SageMath
E = EllipticCurve("df1")
E.isogeny_class()
Elliptic curves in class 143325df
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 143325.de2 | 143325df1 | \([0, 0, 1, -80850, 8864406]\) | \(-43614208/91\) | \(-121948703296875\) | \([]\) | \(497664\) | \(1.5885\) | \(\Gamma_0(N)\)-optimal |
| 143325.de3 | 143325df2 | \([0, 0, 1, 139650, 43979031]\) | \(224755712/753571\) | \(-1009857212001421875\) | \([]\) | \(1492992\) | \(2.1378\) | |
| 143325.de1 | 143325df3 | \([0, 0, 1, -1293600, -1395720594]\) | \(-178643795968/524596891\) | \(-703010006714528296875\) | \([]\) | \(4478976\) | \(2.6871\) |
Rank
sage: E.rank()
The elliptic curves in class 143325df have rank \(0\).
Complex multiplication
The elliptic curves in class 143325df do not have complex multiplication.Modular form 143325.2.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.
\(\left(\begin{array}{rrr} 1 & 3 & 9 \\ 3 & 1 & 3 \\ 9 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
