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SageMath
E = EllipticCurve("c1")
E.isogeny_class()
Elliptic curves in class 42978.c
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 42978.c1 | 42978b4 | \([1, 1, 0, -16059278499, -783322981022457]\) | \(458038307459437803276572539343003833/86000447460798\) | \(86000447460798\) | \([2]\) | \(40919040\) | \(3.9675\) | |
| 42978.c2 | 42978b2 | \([1, 1, 0, -1003704909, -12239735151735]\) | \(111825759760338976846738658338393/1532291201797601099556\) | \(1532291201797601099556\) | \([2, 2]\) | \(20459520\) | \(3.6209\) | |
| 42978.c3 | 42978b3 | \([1, 1, 0, -1002813239, -12262566362085]\) | \(-111527993597885114164012178708473/413980765601504764798430334\) | \(-413980765601504764798430334\) | \([2]\) | \(40919040\) | \(3.9675\) | |
| 42978.c4 | 42978b1 | \([1, 1, 0, -62787289, -190908660587]\) | \(27374041292637614212993237273/101051566314387812377488\) | \(101051566314387812377488\) | \([2]\) | \(10229760\) | \(3.2743\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 42978.c have rank \(1\).
Complex multiplication
The elliptic curves in class 42978.c do not have complex multiplication.Modular form 42978.2.a.c
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 LMFDB numbering.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
