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SageMath
E = EllipticCurve("eu1")
E.isogeny_class()
Elliptic curves in class 364650.eu
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 364650.eu1 | 364650eu4 | \([1, 1, 1, -37400088, -88050924219]\) | \(370272585841418777690809/8011360500\) | \(125177507812500\) | \([2]\) | \(21233664\) | \(2.6837\) | |
| 364650.eu2 | 364650eu2 | \([1, 1, 1, -2337588, -1376424219]\) | \(90408154723558130809/13296962250000\) | \(207765035156250000\) | \([2, 2]\) | \(10616832\) | \(2.3372\) | |
| 364650.eu3 | 364650eu3 | \([1, 1, 1, -2123088, -1638972219]\) | \(-67734150096518559289/34990731445312500\) | \(-546730178833007812500\) | \([2]\) | \(21233664\) | \(2.6837\) | |
| 364650.eu4 | 364650eu1 | \([1, 1, 1, -159588, -17352219]\) | \(28767661004375929/8386833312000\) | \(131044270500000000\) | \([2]\) | \(5308416\) | \(1.9906\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 364650.eu have rank \(1\).
Complex multiplication
The elliptic curves in class 364650.eu do not have complex multiplication.Modular form 364650.2.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 LMFDB 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 LMFDB labels.
