Show commands:
SageMath
E = EllipticCurve("bc1")
E.isogeny_class()
Elliptic curves in class 31200bc
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 31200.l3 | 31200bc1 | \([0, -1, 0, -40041158, -97503029688]\) | \(7099759044484031233216/577161945398025\) | \(577161945398025000000\) | \([2, 2]\) | \(2580480\) | \(3.0289\) | \(\Gamma_0(N)\)-optimal |
| 31200.l4 | 31200bc2 | \([0, -1, 0, -37307408, -111390479688]\) | \(-717825640026599866952/254764560814329735\) | \(-2038116486514637880000000\) | \([2]\) | \(5160960\) | \(3.3754\) | |
| 31200.l2 | 31200bc3 | \([0, -1, 0, -42787408, -83359842188]\) | \(1082883335268084577352/251301565117746585\) | \(2010412520941972680000000\) | \([2]\) | \(5160960\) | \(3.3754\) | |
| 31200.l1 | 31200bc4 | \([0, -1, 0, -640646033, -6241090296063]\) | \(454357982636417669333824/3003024375\) | \(192193560000000000\) | \([2]\) | \(5160960\) | \(3.3754\) |
Rank
sage: E.rank()
The elliptic curves in class 31200bc have rank \(0\).
Complex multiplication
The elliptic curves in class 31200bc do not have complex multiplication.Modular form 31200.2.a.bc
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.
