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SageMath
E = EllipticCurve("eu1")
E.isogeny_class()
Elliptic curves in class 296208eu
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 296208.eu5 | 296208eu1 | \([0, 0, 0, -4948779, -3209650598]\) | \(2533811507137/625016832\) | \(3306247039442346835968\) | \([2]\) | \(11796480\) | \(2.8396\) | \(\Gamma_0(N)\)-optimal |
| 296208.eu4 | 296208eu2 | \([0, 0, 0, -27251499, 52087713370]\) | \(423108074414017/23284318464\) | \(123170617246725241307136\) | \([2, 2]\) | \(23592960\) | \(3.1862\) | |
| 296208.eu2 | 296208eu3 | \([0, 0, 0, -430094379, 3433148005210]\) | \(1663303207415737537/5483698704\) | \(29007958949326078672896\) | \([2, 2]\) | \(47185920\) | \(3.5328\) | |
| 296208.eu6 | 296208eu4 | \([0, 0, 0, 18747861, 210058715482]\) | \(137763859017023/3683199928848\) | \(-19483585460347990301540352\) | \([4]\) | \(47185920\) | \(3.5328\) | |
| 296208.eu1 | 296208eu5 | \([0, 0, 0, -6881504619, 219721837429402]\) | \(6812873765474836663297/74052\) | \(391724179621429248\) | \([2]\) | \(94371840\) | \(3.8793\) | |
| 296208.eu3 | 296208eu6 | \([0, 0, 0, -424170219, 3532317258778]\) | \(-1595514095015181697/95635786040388\) | \(-505899230663878419791757312\) | \([2]\) | \(94371840\) | \(3.8793\) |
Rank
sage: E.rank()
The elliptic curves in class 296208eu have rank \(1\).
Complex multiplication
The elliptic curves in class 296208eu do not have complex multiplication.Modular form 296208.2.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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 2 & 2 \\ 4 & 2 & 4 & 1 & 8 & 8 \\ 8 & 4 & 2 & 8 & 1 & 4 \\ 8 & 4 & 2 & 8 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
