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SageMath
E = EllipticCurve("fj1")
E.isogeny_class()
Elliptic curves in class 388080fj
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 388080.fj3 | 388080fj1 | \([0, 0, 0, -207123, 172410322]\) | \(-75526045083/943250000\) | \(-12272661789696000000\) | \([2]\) | \(7962624\) | \(2.3446\) | \(\Gamma_0(N)\)-optimal |
| 388080.fj2 | 388080fj2 | \([0, 0, 0, -6087123, 5761938322]\) | \(1917114236485083/7117764500\) | \(92609505865046016000\) | \([2]\) | \(15925248\) | \(2.6912\) | |
| 388080.fj4 | 388080fj3 | \([0, 0, 0, 1850877, -4473044478]\) | \(73929353373/954060800\) | \(-9049311389213628825600\) | \([2]\) | \(23887872\) | \(2.8939\) | |
| 388080.fj1 | 388080fj4 | \([0, 0, 0, -32017923, -65213350398]\) | \(382704614800227/27778076480\) | \(263476356916573061775360\) | \([2]\) | \(47775744\) | \(3.2405\) |
Rank
sage: E.rank()
The elliptic curves in class 388080fj have rank \(0\).
Complex multiplication
The elliptic curves in class 388080fj do not have complex multiplication.Modular form 388080.2.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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
