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SageMath
E = EllipticCurve("r1")
E.isogeny_class()
Elliptic curves in class 126960r
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
126960.cx3 | 126960r1 | \([0, 1, 0, -61540, -5894020]\) | \(680136784/345\) | \(13074529716480\) | \([2]\) | \(405504\) | \(1.4688\) | \(\Gamma_0(N)\)-optimal |
126960.cx2 | 126960r2 | \([0, 1, 0, -72120, -3739932]\) | \(273671716/119025\) | \(18042851008742400\) | \([2, 2]\) | \(811008\) | \(1.8154\) | |
126960.cx4 | 126960r3 | \([0, 1, 0, 245280, -27481452]\) | \(5382838942/4197615\) | \(-1272622424483297280\) | \([2]\) | \(1622016\) | \(2.1620\) | |
126960.cx1 | 126960r4 | \([0, 1, 0, -558800, 158032500]\) | \(63649751618/1164375\) | \(353012302344960000\) | \([4]\) | \(1622016\) | \(2.1620\) |
Rank
sage: E.rank()
The elliptic curves in class 126960r have rank \(0\).
Complex multiplication
The elliptic curves in class 126960r do not have complex multiplication.Modular form 126960.2.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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.