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SageMath
E = EllipticCurve("fn1")
E.isogeny_class()
Elliptic curves in class 414960fn
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
414960.fn4 | 414960fn1 | \([0, 1, 0, 13944, -1106700]\) | \(73197245859191/172623360000\) | \(-707065282560000\) | \([2]\) | \(1474560\) | \(1.5333\) | \(\Gamma_0(N)\)-optimal* |
414960.fn3 | 414960fn2 | \([0, 1, 0, -114056, -12319500]\) | \(40061018056412809/7275103617600\) | \(29798824417689600\) | \([2, 2]\) | \(2949120\) | \(1.8799\) | \(\Gamma_0(N)\)-optimal* |
414960.fn2 | 414960fn3 | \([0, 1, 0, -539656, 141066740]\) | \(4243415895694547209/351514682293320\) | \(1439804138673438720\) | \([2]\) | \(5898240\) | \(2.2265\) | \(\Gamma_0(N)\)-optimal* |
414960.fn1 | 414960fn4 | \([0, 1, 0, -1736456, -881276940]\) | \(141369383441705190409/6345626621880\) | \(25991686643220480\) | \([2]\) | \(5898240\) | \(2.2265\) |
Rank
sage: E.rank()
The elliptic curves in class 414960fn have rank \(0\).
Complex multiplication
The elliptic curves in class 414960fn do not have complex multiplication.Modular form 414960.2.a.fn
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.