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SageMath
E = EllipticCurve("il1")
E.isogeny_class()
Elliptic curves in class 122304il
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 122304.fm3 | 122304il1 | \([0, 1, 0, -7709, 257907]\) | \(420616192/117\) | \(14095291392\) | \([2]\) | \(147456\) | \(0.92921\) | \(\Gamma_0(N)\)-optimal |
| 122304.fm2 | 122304il2 | \([0, 1, 0, -8689, 187151]\) | \(37642192/13689\) | \(26386385485824\) | \([2, 2]\) | \(294912\) | \(1.2758\) | |
| 122304.fm4 | 122304il3 | \([0, 1, 0, 26591, 1351391]\) | \(269676572/257049\) | \(-1981910732046336\) | \([2]\) | \(589824\) | \(1.6224\) | |
| 122304.fm1 | 122304il4 | \([0, 1, 0, -59649, -5489793]\) | \(3044193988/85293\) | \(657629915185152\) | \([2]\) | \(589824\) | \(1.6224\) |
Rank
sage: E.rank()
The elliptic curves in class 122304il have rank \(1\).
Complex multiplication
The elliptic curves in class 122304il do not have complex multiplication.Modular form 122304.2.a.il
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.
