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SageMath
E = EllipticCurve("fi1")
E.isogeny_class()
Elliptic curves in class 433200fi
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
433200.fi4 | 433200fi1 | \([0, -1, 0, 530883592, -28257068876688]\) | \(5495662324535111/117739817533440\) | \(-354507100456955632680960000000\) | \([2]\) | \(464486400\) | \(4.3503\) | \(\Gamma_0(N)\)-optimal* |
433200.fi3 | 433200fi2 | \([0, -1, 0, -11298364408, -437832951628688]\) | \(52974743974734147769/3152005008998400\) | \(9490486564143529957785600000000\) | \([2, 2]\) | \(928972800\) | \(4.6969\) | \(\Gamma_0(N)\)-optimal* |
433200.fi2 | 433200fi3 | \([0, -1, 0, -33755452408, 1843088562355312]\) | \(1412712966892699019449/330160465517040000\) | \(994092158183633107983360000000000\) | \([2]\) | \(1857945600\) | \(5.0435\) | \(\Gamma_0(N)\)-optimal* |
433200.fi1 | 433200fi4 | \([0, -1, 0, -178109244408, -28931800229708688]\) | \(207530301091125281552569/805586668007040\) | \(2425570209167725504143360000000\) | \([2]\) | \(1857945600\) | \(5.0435\) |
Rank
sage: E.rank()
The elliptic curves in class 433200fi have rank \(1\).
Complex multiplication
The elliptic curves in class 433200fi do not have complex multiplication.Modular form 433200.2.a.fi
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.