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SageMath
E = EllipticCurve("rm1")
E.isogeny_class()
Elliptic curves in class 310464.rm
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
310464.rm1 | 310464rm2 | \([0, 0, 0, -210128268, 1172099517680]\) | \(45637459887836881/13417633152\) | \(301669639375563886952448\) | \([2]\) | \(99090432\) | \(3.4843\) | |
310464.rm2 | 310464rm1 | \([0, 0, 0, -11431308, 23233694960]\) | \(-7347774183121/6119866368\) | \(-137593408565206981804032\) | \([2]\) | \(49545216\) | \(3.1377\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 310464.rm have rank \(1\).
Complex multiplication
The elliptic curves in class 310464.rm do not have complex multiplication.Modular form 310464.2.a.rm
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.