Show commands:
SageMath
E = EllipticCurve("bh1")
E.isogeny_class()
Elliptic curves in class 159936.bh
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 159936.bh1 | 159936ip4 | \([0, -1, 0, -15927809, -24461764095]\) | \(14489843500598257/6246072\) | \(192634978232696832\) | \([2]\) | \(7077888\) | \(2.6592\) | |
| 159936.bh2 | 159936ip3 | \([0, -1, 0, -2129409, 632909313]\) | \(34623662831857/14438442312\) | \(445295702721033142272\) | \([2]\) | \(7077888\) | \(2.6592\) | |
| 159936.bh3 | 159936ip2 | \([0, -1, 0, -1000449, -377961471]\) | \(3590714269297/73410624\) | \(2264055546636140544\) | \([2, 2]\) | \(3538944\) | \(2.3127\) | |
| 159936.bh4 | 159936ip1 | \([0, -1, 0, 3071, -17697791]\) | \(103823/4386816\) | \(-135293702133252096\) | \([2]\) | \(1769472\) | \(1.9661\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 159936.bh have rank \(2\).
Complex multiplication
The elliptic curves in class 159936.bh do not have complex multiplication.Modular form 159936.2.a.bh
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.
