Show commands:
SageMath
E = EllipticCurve("a1")
E.isogeny_class()
Elliptic curves in class 16473a
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 16473.e3 | 16473a1 | \([1, 1, 0, -439, -3248]\) | \(389017/57\) | \(1375841433\) | \([2]\) | \(6912\) | \(0.47845\) | \(\Gamma_0(N)\)-optimal |
| 16473.e2 | 16473a2 | \([1, 1, 0, -1884, 27675]\) | \(30664297/3249\) | \(78422961681\) | \([2, 2]\) | \(13824\) | \(0.82502\) | |
| 16473.e1 | 16473a3 | \([1, 1, 0, -29339, 1922070]\) | \(115714886617/1539\) | \(37147718691\) | \([2]\) | \(27648\) | \(1.1716\) | |
| 16473.e4 | 16473a4 | \([1, 1, 0, 2451, 141252]\) | \(67419143/390963\) | \(-9436896388947\) | \([2]\) | \(27648\) | \(1.1716\) |
Rank
sage: E.rank()
The elliptic curves in class 16473a have rank \(1\).
Complex multiplication
The elliptic curves in class 16473a do not have complex multiplication.Modular form 16473.2.a.a
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
