Show commands:
SageMath
E = EllipticCurve("bx1")
E.isogeny_class()
Elliptic curves in class 75712.bx
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 75712.bx1 | 75712y4 | \([0, 0, 0, -639589964, 6223771098480]\) | \(22868021811807457713/8953460393696\) | \(11328983717494232465801216\) | \([2]\) | \(23224320\) | \(3.7722\) | |
| 75712.bx2 | 75712y3 | \([0, 0, 0, -338472524, -2350626226832]\) | \(3389174547561866673/74853681183008\) | \(94713786405296186712915968\) | \([2]\) | \(23224320\) | \(3.7722\) | |
| 75712.bx3 | 75712y2 | \([0, 0, 0, -46007884, 65950600560]\) | \(8511781274893233/3440817243136\) | \(4353731496908956113043456\) | \([2, 2]\) | \(11612160\) | \(3.4256\) | |
| 75712.bx4 | 75712y1 | \([0, 0, 0, 9370036, 7493668208]\) | \(71903073502287/60782804992\) | \(-76909639153911209132032\) | \([2]\) | \(5806080\) | \(3.0791\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 75712.bx have rank \(0\).
Complex multiplication
The elliptic curves in class 75712.bx do not have complex multiplication.Modular form 75712.2.a.bx
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.
