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SageMath
E = EllipticCurve("eu1")
E.isogeny_class()
Elliptic curves in class 462400eu
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 462400.eu3 | 462400eu1 | \([0, 0, 0, -4132700, 3232754000]\) | \(1263257424/425\) | \(2626167507200000000\) | \([2]\) | \(7077888\) | \(2.5066\) | \(\Gamma_0(N)\)-optimal |
| 462400.eu2 | 462400eu2 | \([0, 0, 0, -4710700, 2269806000]\) | \(467720676/180625\) | \(4464484762240000000000\) | \([2, 2]\) | \(14155776\) | \(2.8532\) | |
| 462400.eu4 | 462400eu3 | \([0, 0, 0, 14941300, 16301334000]\) | \(7462174302/6640625\) | \(-328270938400000000000000\) | \([2]\) | \(28311552\) | \(3.1998\) | |
| 462400.eu1 | 462400eu4 | \([0, 0, 0, -33610700, -73390394000]\) | \(84944038338/2088025\) | \(103218887702988800000000\) | \([2]\) | \(28311552\) | \(3.1998\) |
Rank
sage: E.rank()
The elliptic curves in class 462400eu have rank \(1\).
Complex multiplication
The elliptic curves in class 462400eu do not have complex multiplication.Modular form 462400.2.a.eu
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.
