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SageMath
E = EllipticCurve("by1")
E.isogeny_class()
Elliptic curves in class 31200.by
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 31200.by1 | 31200bv4 | \([0, 1, 0, -640646033, 6241090296063]\) | \(454357982636417669333824/3003024375\) | \(192193560000000000\) | \([2]\) | \(5160960\) | \(3.3754\) | |
| 31200.by2 | 31200bv3 | \([0, 1, 0, -42787408, 83359842188]\) | \(1082883335268084577352/251301565117746585\) | \(2010412520941972680000000\) | \([2]\) | \(5160960\) | \(3.3754\) | |
| 31200.by3 | 31200bv1 | \([0, 1, 0, -40041158, 97503029688]\) | \(7099759044484031233216/577161945398025\) | \(577161945398025000000\) | \([2, 2]\) | \(2580480\) | \(3.0289\) | \(\Gamma_0(N)\)-optimal |
| 31200.by4 | 31200bv2 | \([0, 1, 0, -37307408, 111390479688]\) | \(-717825640026599866952/254764560814329735\) | \(-2038116486514637880000000\) | \([2]\) | \(5160960\) | \(3.3754\) |
Rank
sage: E.rank()
The elliptic curves in class 31200.by have rank \(1\).
Complex multiplication
The elliptic curves in class 31200.by do not have complex multiplication.Modular form 31200.2.a.by
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 LMFDB numbering.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
