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SageMath
E = EllipticCurve("ml1")
E.isogeny_class()
Elliptic curves in class 187200ml
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 187200.hz3 | 187200ml1 | \([0, 0, 0, -58575, 5456500]\) | \(30488290624/195\) | \(142155000000\) | \([2]\) | \(294912\) | \(1.3247\) | \(\Gamma_0(N)\)-optimal |
| 187200.hz2 | 187200ml2 | \([0, 0, 0, -59700, 5236000]\) | \(504358336/38025\) | \(1774094400000000\) | \([2, 2]\) | \(589824\) | \(1.6712\) | |
| 187200.hz4 | 187200ml3 | \([0, 0, 0, 57300, 23254000]\) | \(55742968/658125\) | \(-245643840000000000\) | \([2]\) | \(1179648\) | \(2.0178\) | |
| 187200.hz1 | 187200ml4 | \([0, 0, 0, -194700, -26894000]\) | \(2186875592/428415\) | \(159905041920000000\) | \([2]\) | \(1179648\) | \(2.0178\) |
Rank
sage: E.rank()
The elliptic curves in class 187200ml have rank \(2\).
Complex multiplication
The elliptic curves in class 187200ml do not have complex multiplication.Modular form 187200.2.a.ml
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.
