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SageMath
E = EllipticCurve("j1")
E.isogeny_class()
Elliptic curves in class 31824j
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
31824.r1 | 31824j1 | \([0, 0, 0, -128595, 17687842]\) | \(315042014258500/1262881737\) | \(942736165143552\) | \([2]\) | \(122880\) | \(1.7299\) | \(\Gamma_0(N)\)-optimal |
31824.r2 | 31824j2 | \([0, 0, 0, -67755, 34467514]\) | \(-23040414103250/330419182041\) | \(-493313195433756672\) | \([2]\) | \(245760\) | \(2.0765\) |
Rank
sage: E.rank()
The elliptic curves in class 31824j have rank \(1\).
Complex multiplication
The elliptic curves in class 31824j do not have complex multiplication.Modular form 31824.2.a.j
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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.