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SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 219351e
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 219351.c4 | 219351e1 | \([1, 1, 1, -455916869, -3881891248558]\) | \(-434197349785010750259313/18399506773223217807\) | \(-444119364304642772218491183\) | \([2]\) | \(132120576\) | \(3.8795\) | \(\Gamma_0(N)\)-optimal |
| 219351.c3 | 219351e2 | \([1, 1, 1, -7367307074, -243397501080874]\) | \(1832130900601560534748842433/3093995404133997561\) | \(74681527552967251374469209\) | \([2, 2]\) | \(264241152\) | \(4.2261\) | |
| 219351.c2 | 219351e3 | \([1, 1, 1, -7439992019, -238349822390404]\) | \(1886894388313703307822840913/75215867942322411289821\) | \(1815528202352695222754433385149\) | \([2]\) | \(528482304\) | \(4.5727\) | |
| 219351.c1 | 219351e4 | \([1, 1, 1, -117876865409, -15577305981235468]\) | \(7504399044296074448738193191473/21401456964723\) | \(516579144186531978387\) | \([2]\) | \(528482304\) | \(4.5727\) |
Rank
sage: E.rank()
The elliptic curves in class 219351e have rank \(0\).
Complex multiplication
The elliptic curves in class 219351e do not have complex multiplication.Modular form 219351.2.a.e
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.
