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SageMath
E = EllipticCurve("a1")
E.isogeny_class()
Elliptic curves in class 1610.a
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
1610.a1 | 1610a2 | \([1, -1, 0, -15610, -746700]\) | \(420676324562824569/56350000000\) | \(56350000000\) | \([2]\) | \(2688\) | \(1.0820\) | |
1610.a2 | 1610a1 | \([1, -1, 0, -890, -13644]\) | \(-78013216986489/37918720000\) | \(-37918720000\) | \([2]\) | \(1344\) | \(0.73538\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 1610.a have rank \(1\).
Complex multiplication
The elliptic curves in class 1610.a do not have complex multiplication.Modular form 1610.2.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 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.