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SageMath
E = EllipticCurve("j1")
E.isogeny_class()
Elliptic curves in class 18720j
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 18720.u3 | 18720j1 | \([0, 0, 0, -146433, 12204632]\) | \(7442744143086784/2927948765625\) | \(136606377609000000\) | \([2, 2]\) | \(196608\) | \(1.9865\) | \(\Gamma_0(N)\)-optimal |
| 18720.u2 | 18720j2 | \([0, 0, 0, -1057683, -410068618]\) | \(350584567631475848/8259273550125\) | \(3082757334037056000\) | \([2]\) | \(393216\) | \(2.3331\) | |
| 18720.u1 | 18720j3 | \([0, 0, 0, -2047683, 1127477882]\) | \(2543984126301795848/909361981125\) | \(339417540730944000\) | \([2]\) | \(393216\) | \(2.3331\) | |
| 18720.u4 | 18720j4 | \([0, 0, 0, 469572, 88096448]\) | \(3834800837445824/3342041015625\) | \(-9979281000000000000\) | \([2]\) | \(393216\) | \(2.3331\) |
Rank
sage: E.rank()
The elliptic curves in class 18720j have rank \(1\).
Complex multiplication
The elliptic curves in class 18720j do not have complex multiplication.Modular form 18720.2.a.j
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 2 & 2 \\ 2 & 1 & 4 & 4 \\ 2 & 4 & 1 & 4 \\ 2 & 4 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
