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SageMath
E = EllipticCurve("fj1")
E.isogeny_class()
Elliptic curves in class 177870fj
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 177870.eb3 | 177870fj1 | \([1, 0, 1, -2970553, -1711362292]\) | \(13908844989649/1980372240\) | \(412753895722984329360\) | \([2]\) | \(8847360\) | \(2.6815\) | \(\Gamma_0(N)\)-optimal |
| 177870.eb2 | 177870fj2 | \([1, 0, 1, -12575533, 15458499956]\) | \(1055257664218129/115307784900\) | \(24032722971648454856100\) | \([2, 2]\) | \(17694720\) | \(3.0281\) | |
| 177870.eb1 | 177870fj3 | \([1, 0, 1, -195603763, 1052935718888]\) | \(3971101377248209009/56495958750\) | \(11775022188084965328750\) | \([2]\) | \(35389440\) | \(3.3747\) | |
| 177870.eb4 | 177870fj4 | \([1, 0, 1, 16773017, 76890884816]\) | \(2503876820718671/13702874328990\) | \(-2855985681708554611480110\) | \([2]\) | \(35389440\) | \(3.3747\) |
Rank
sage: E.rank()
The elliptic curves in class 177870fj have rank \(1\).
Complex multiplication
The elliptic curves in class 177870fj do not have complex multiplication.Modular form 177870.2.a.fj
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.
