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SageMath
E = EllipticCurve("s1")
E.isogeny_class()
Elliptic curves in class 304920s
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 304920.s5 | 304920s1 | \([0, 0, 0, 2509782, 5040154933]\) | \(84611246065664/580054565475\) | \(-11985950265330812324400\) | \([2]\) | \(15728640\) | \(2.9182\) | \(\Gamma_0(N)\)-optimal |
| 304920.s4 | 304920s2 | \([0, 0, 0, -33214863, 67208182162]\) | \(12257375872392016/1191317675625\) | \(393868446457144206240000\) | \([2, 2]\) | \(31457280\) | \(3.2648\) | |
| 304920.s2 | 304920s3 | \([0, 0, 0, -518364363, 4542518259862]\) | \(11647843478225136004/128410942275\) | \(169818745668884981222400\) | \([2]\) | \(62914560\) | \(3.6113\) | |
| 304920.s3 | 304920s4 | \([0, 0, 0, -119659683, -429348152882]\) | \(143279368983686884/22699269140625\) | \(30018948111183200400000000\) | \([2, 2]\) | \(62914560\) | \(3.6113\) | |
| 304920.s6 | 304920s5 | \([0, 0, 0, 212398197, -2387891940698]\) | \(400647648358480318/1163177490234375\) | \(-3076518852402187500000000000\) | \([2]\) | \(125829120\) | \(3.9579\) | |
| 304920.s1 | 304920s6 | \([0, 0, 0, -1834834683, -30250409807882]\) | \(258286045443018193442/8440380939375\) | \(22324186376931212040960000\) | \([2]\) | \(125829120\) | \(3.9579\) |
Rank
sage: E.rank()
The elliptic curves in class 304920s have rank \(0\).
Complex multiplication
The elliptic curves in class 304920s do not have complex multiplication.Modular form 304920.2.a.s
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 8 & 8 \\ 4 & 2 & 4 & 1 & 2 & 2 \\ 8 & 4 & 8 & 2 & 1 & 4 \\ 8 & 4 & 8 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
