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SageMath
E = EllipticCurve("id1")
E.isogeny_class()
Elliptic curves in class 433200.id
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
433200.id1 | 433200id4 | \([0, 1, 0, -89675408, -326412040812]\) | \(26487576322129/44531250\) | \(134080760850000000000000\) | \([2]\) | \(53084160\) | \(3.3330\) | |
433200.id2 | 433200id2 | \([0, 1, 0, -7367408, -1624672812]\) | \(14688124849/8122500\) | \(24456330779040000000000\) | \([2, 2]\) | \(26542080\) | \(2.9864\) | |
433200.id3 | 433200id1 | \([0, 1, 0, -4479408, 3625711188]\) | \(3301293169/22800\) | \(68649349555200000000\) | \([2]\) | \(13271040\) | \(2.6399\) | \(\Gamma_0(N)\)-optimal* |
433200.id4 | 433200id3 | \([0, 1, 0, 28732592, -12815672812]\) | \(871257511151/527800050\) | \(-1589172374022019200000000\) | \([2]\) | \(53084160\) | \(3.3330\) |
Rank
sage: E.rank()
The elliptic curves in class 433200.id have rank \(0\).
Complex multiplication
The elliptic curves in class 433200.id do not have complex multiplication.Modular form 433200.2.a.id
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.