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SageMath
E = EllipticCurve("gs1")
E.isogeny_class()
Elliptic curves in class 259350.gs
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
259350.gs1 | 259350gs3 | \([1, 0, 0, -2713213, 1719887417]\) | \(141369383441705190409/6345626621880\) | \(99150415966875000\) | \([2]\) | \(5898240\) | \(2.3380\) | |
259350.gs2 | 259350gs4 | \([1, 0, 0, -843213, -275942583]\) | \(4243415895694547209/351514682293320\) | \(5492416910833125000\) | \([2]\) | \(5898240\) | \(2.3380\) | |
259350.gs3 | 259350gs2 | \([1, 0, 0, -178213, 23972417]\) | \(40061018056412809/7275103617600\) | \(113673494025000000\) | \([2, 2]\) | \(2949120\) | \(1.9915\) | |
259350.gs4 | 259350gs1 | \([1, 0, 0, 21787, 2172417]\) | \(73197245859191/172623360000\) | \(-2697240000000000\) | \([2]\) | \(1474560\) | \(1.6449\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 259350.gs have rank \(0\).
Complex multiplication
The elliptic curves in class 259350.gs do not have complex multiplication.Modular form 259350.2.a.gs
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.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.