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SageMath
E = EllipticCurve("ca1")
E.isogeny_class()
Elliptic curves in class 343824ca
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 343824.ca4 | 343824ca1 | \([0, 1, 0, -1004596632, 12216145084308]\) | \(27374041292637614212993237273/101051566314387812377488\) | \(413907215623732479498190848\) | \([2]\) | \(245514240\) | \(3.9675\) | \(\Gamma_0(N)\)-optimal |
| 343824.ca2 | 343824ca2 | \([0, 1, 0, -16059278552, 783310931153940]\) | \(111825759760338976846738658338393/1532291201797601099556\) | \(6276264762562974103781376\) | \([2, 2]\) | \(491028480\) | \(4.3140\) | |
| 343824.ca1 | 343824ca3 | \([0, 1, 0, -256948455992, 50132156888525268]\) | \(458038307459437803276572539343003833/86000447460798\) | \(352257832799428608\) | \([2]\) | \(982056960\) | \(4.6606\) | |
| 343824.ca3 | 343824ca4 | \([0, 1, 0, -16045011832, 784772157149780]\) | \(-111527993597885114164012178708473/413980765601504764798430334\) | \(-1695665215903763516614370648064\) | \([2]\) | \(982056960\) | \(4.6606\) |
Rank
sage: E.rank()
The elliptic curves in class 343824ca have rank \(0\).
Complex multiplication
The elliptic curves in class 343824ca do not have complex multiplication.Modular form 343824.2.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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
