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SageMath
E = EllipticCurve("bj1")
E.isogeny_class()
Elliptic curves in class 97104bj
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 97104.bb5 | 97104bj1 | \([0, -1, 0, -18592, -2148608]\) | \(-7189057/16128\) | \(-1594534759759872\) | \([2]\) | \(491520\) | \(1.6057\) | \(\Gamma_0(N)\)-optimal |
| 97104.bb4 | 97104bj2 | \([0, -1, 0, -388512, -93000960]\) | \(65597103937/63504\) | \(6278480616554496\) | \([2, 2]\) | \(983040\) | \(1.9523\) | |
| 97104.bb3 | 97104bj3 | \([0, -1, 0, -480992, -45281280]\) | \(124475734657/63011844\) | \(6229822391776198656\) | \([2, 2]\) | \(1966080\) | \(2.2989\) | |
| 97104.bb1 | 97104bj4 | \([0, -1, 0, -6214752, -5961189888]\) | \(268498407453697/252\) | \(24914605621248\) | \([2]\) | \(1966080\) | \(2.2989\) | |
| 97104.bb6 | 97104bj5 | \([0, -1, 0, 1784768, -351612032]\) | \(6359387729183/4218578658\) | \(-417080252167792238592\) | \([2]\) | \(3932160\) | \(2.6454\) | |
| 97104.bb2 | 97104bj6 | \([0, -1, 0, -4226432, 3313629312]\) | \(84448510979617/933897762\) | \(92332120757127487488\) | \([2]\) | \(3932160\) | \(2.6454\) |
Rank
sage: E.rank()
The elliptic curves in class 97104bj have rank \(0\).
Complex multiplication
The elliptic curves in class 97104bj do not have complex multiplication.Modular form 97104.2.a.bj
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 2 & 2 \\ 4 & 2 & 4 & 1 & 8 & 8 \\ 8 & 4 & 2 & 8 & 1 & 4 \\ 8 & 4 & 2 & 8 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
