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SageMath
E = EllipticCurve("ep1")
E.isogeny_class()
Elliptic curves in class 121680ep
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 121680.eg6 | 121680ep1 | \([0, 0, 0, 364533, 33926074]\) | \(371694959/249600\) | \(-3597428501485977600\) | \([2]\) | \(2064384\) | \(2.2493\) | \(\Gamma_0(N)\)-optimal |
| 121680.eg5 | 121680ep2 | \([0, 0, 0, -1582347, 281958586]\) | \(30400540561/15210000\) | \(219218299309301760000\) | \([2, 2]\) | \(4128768\) | \(2.5959\) | |
| 121680.eg3 | 121680ep3 | \([0, 0, 0, -13750347, -19427767814]\) | \(19948814692561/231344100\) | \(3334310332494479769600\) | \([2, 2]\) | \(8257536\) | \(2.9424\) | |
| 121680.eg2 | 121680ep4 | \([0, 0, 0, -20564427, 35865765754]\) | \(66730743078481/60937500\) | \(878278442745600000000\) | \([4]\) | \(8257536\) | \(2.9424\) | |
| 121680.eg4 | 121680ep5 | \([0, 0, 0, -2799147, -49523855654]\) | \(-168288035761/73415764890\) | \(-1058124860070831518883840\) | \([2]\) | \(16515072\) | \(3.2890\) | |
| 121680.eg1 | 121680ep6 | \([0, 0, 0, -219389547, -1250754169574]\) | \(81025909800741361/11088090\) | \(159810140196480983040\) | \([2]\) | \(16515072\) | \(3.2890\) |
Rank
sage: E.rank()
The elliptic curves in class 121680ep have rank \(0\).
Complex multiplication
The elliptic curves in class 121680ep do not have complex multiplication.Modular form 121680.2.a.ep
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.
