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SageMath
E = EllipticCurve("r1")
E.isogeny_class()
Elliptic curves in class 31824r
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
31824.t2 | 31824r1 | \([0, 0, 0, -4755, -119662]\) | \(3981876625/232713\) | \(694877294592\) | \([2]\) | \(32768\) | \(1.0258\) | \(\Gamma_0(N)\)-optimal |
31824.t1 | 31824r2 | \([0, 0, 0, -14115, 496226]\) | \(104154702625/24649677\) | \(73603541127168\) | \([2]\) | \(65536\) | \(1.3724\) |
Rank
sage: E.rank()
The elliptic curves in class 31824r have rank \(1\).
Complex multiplication
The elliptic curves in class 31824r do not have complex multiplication.Modular form 31824.2.a.r
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.