Show commands:
SageMath
E = EllipticCurve("a1")
E.isogeny_class()
Elliptic curves in class 6253a
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 6253.b3 | 6253a1 | \([0, 1, 1, -563, 4918]\) | \(4096000/37\) | \(178591933\) | \([]\) | \(1440\) | \(0.40594\) | \(\Gamma_0(N)\)-optimal |
| 6253.b2 | 6253a2 | \([0, 1, 1, -3943, -93609]\) | \(1404928000/50653\) | \(244492356277\) | \([]\) | \(4320\) | \(0.95525\) | |
| 6253.b1 | 6253a3 | \([0, 1, 1, -316593, -68670260]\) | \(727057727488000/37\) | \(178591933\) | \([]\) | \(12960\) | \(1.5046\) |
Rank
sage: E.rank()
The elliptic curves in class 6253a have rank \(1\).
Complex multiplication
The elliptic curves in class 6253a do not have complex multiplication.Modular form 6253.2.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.
\(\left(\begin{array}{rrr} 1 & 3 & 9 \\ 3 & 1 & 3 \\ 9 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
