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SageMath
E = EllipticCurve("ei1")
E.isogeny_class()
Elliptic curves in class 414960ei
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
414960.ei4 | 414960ei1 | \([0, 1, 0, -1107496, 126785204]\) | \(36676733979624816169/19519718400000000\) | \(79952766566400000000\) | \([2]\) | \(14155776\) | \(2.5104\) | \(\Gamma_0(N)\)-optimal* |
414960.ei2 | 414960ei2 | \([0, 1, 0, -13907496, 19936065204]\) | \(72629093972969564016169/93022316019360000\) | \(381019406415298560000\) | \([2, 2]\) | \(28311552\) | \(2.8569\) | \(\Gamma_0(N)\)-optimal* |
414960.ei1 | 414960ei3 | \([0, 1, 0, -222451496, 1276955879604]\) | \(297214339265273649756432169/488484917902800\) | \(2000834223729868800\) | \([4]\) | \(56623104\) | \(3.2035\) | \(\Gamma_0(N)\)-optimal* |
414960.ei3 | 414960ei4 | \([0, 1, 0, -10163496, 30914970804]\) | \(-28346090452899214800169/84418326220247182800\) | \(-345777464198132460748800\) | \([2]\) | \(56623104\) | \(3.2035\) |
Rank
sage: E.rank()
The elliptic curves in class 414960ei have rank \(0\).
Complex multiplication
The elliptic curves in class 414960ei do not have complex multiplication.Modular form 414960.2.a.ei
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.