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SageMath
E = EllipticCurve("jq1")
E.isogeny_class()
Elliptic curves in class 100800.jq
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 100800.jq1 | 100800fu4 | \([0, 0, 0, -19353900, -32771842000]\) | \(268498407453697/252\) | \(752467968000000\) | \([2]\) | \(3145728\) | \(2.5828\) | |
| 100800.jq2 | 100800fu6 | \([0, 0, 0, -13161900, 18202718000]\) | \(84448510979617/933897762\) | \(2788603774967808000000\) | \([2]\) | \(6291456\) | \(2.9294\) | |
| 100800.jq3 | 100800fu3 | \([0, 0, 0, -1497900, -249730000]\) | \(124475734657/63011844\) | \(188152357994496000000\) | \([2, 2]\) | \(3145728\) | \(2.5828\) | |
| 100800.jq4 | 100800fu2 | \([0, 0, 0, -1209900, -511810000]\) | \(65597103937/63504\) | \(189621927936000000\) | \([2, 2]\) | \(1572864\) | \(2.2363\) | |
| 100800.jq5 | 100800fu1 | \([0, 0, 0, -57900, -11842000]\) | \(-7189057/16128\) | \(-48157949952000000\) | \([2]\) | \(786432\) | \(1.8897\) | \(\Gamma_0(N)\)-optimal |
| 100800.jq6 | 100800fu5 | \([0, 0, 0, 5558100, -1929058000]\) | \(6359387729183/4218578658\) | \(-12596608375529472000000\) | \([2]\) | \(6291456\) | \(2.9294\) |
Rank
sage: E.rank()
The elliptic curves in class 100800.jq have rank \(1\).
Complex multiplication
The elliptic curves in class 100800.jq do not have complex multiplication.Modular form 100800.2.a.jq
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.
\(\left(\begin{array}{rrrrrr} 1 & 8 & 4 & 2 & 4 & 8 \\ 8 & 1 & 2 & 4 & 8 & 4 \\ 4 & 2 & 1 & 2 & 4 & 2 \\ 2 & 4 & 2 & 1 & 2 & 4 \\ 4 & 8 & 4 & 2 & 1 & 8 \\ 8 & 4 & 2 & 4 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
