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SageMath
E = EllipticCurve("ll1")
E.isogeny_class()
Elliptic curves in class 187200ll
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 187200.ej3 | 187200ll1 | \([0, 0, 0, -86700, -63074000]\) | \(-24137569/561600\) | \(-1676928614400000000\) | \([2]\) | \(1769472\) | \(2.1777\) | \(\Gamma_0(N)\)-optimal |
| 187200.ej2 | 187200ll2 | \([0, 0, 0, -2966700, -1958114000]\) | \(967068262369/4928040\) | \(14715048591360000000\) | \([2]\) | \(3538944\) | \(2.5243\) | |
| 187200.ej4 | 187200ll3 | \([0, 0, 0, 777300, 1666654000]\) | \(17394111071/411937500\) | \(-1230038784000000000000\) | \([2]\) | \(5308416\) | \(2.7270\) | |
| 187200.ej1 | 187200ll4 | \([0, 0, 0, -17222700, 26110654000]\) | \(189208196468929/10860320250\) | \(32428742501376000000000\) | \([2]\) | \(10616832\) | \(3.0736\) |
Rank
sage: E.rank()
The elliptic curves in class 187200ll have rank \(1\).
Complex multiplication
The elliptic curves in class 187200ll do not have complex multiplication.Modular form 187200.2.a.ll
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
