Show commands:
SageMath
E = EllipticCurve("hd1")
E.isogeny_class()
Elliptic curves in class 227136.hd
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 227136.hd1 | 227136fr1 | \([0, 1, 0, -1408, 18734]\) | \(1000000/63\) | \(19461693888\) | \([2]\) | \(147456\) | \(0.72536\) | \(\Gamma_0(N)\)-optimal |
| 227136.hd2 | 227136fr2 | \([0, 1, 0, 1127, 81095]\) | \(8000/147\) | \(-2906279620608\) | \([2]\) | \(294912\) | \(1.0719\) |
Rank
sage: E.rank()
The elliptic curves in class 227136.hd have rank \(1\).
Complex multiplication
The elliptic curves in class 227136.hd do not have complex multiplication.Modular form 227136.2.a.hd
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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
