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SageMath
E = EllipticCurve("bh1")
E.isogeny_class()
Elliptic curves in class 123840bh
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
123840.by4 | 123840bh1 | \([0, 0, 0, 852, -87248]\) | \(357911/17415\) | \(-3328058327040\) | \([2]\) | \(180224\) | \(1.0825\) | \(\Gamma_0(N)\)-optimal |
123840.by3 | 123840bh2 | \([0, 0, 0, -25068, -1466192]\) | \(9116230969/416025\) | \(79503615590400\) | \([2, 2]\) | \(360448\) | \(1.4291\) | |
123840.by2 | 123840bh3 | \([0, 0, 0, -68268, 4944688]\) | \(184122897769/51282015\) | \(9800145681776640\) | \([2]\) | \(720896\) | \(1.7757\) | |
123840.by1 | 123840bh4 | \([0, 0, 0, -396588, -96129488]\) | \(36097320816649/80625\) | \(15407677440000\) | \([2]\) | \(720896\) | \(1.7757\) |
Rank
sage: E.rank()
The elliptic curves in class 123840bh have rank \(0\).
Complex multiplication
The elliptic curves in class 123840bh do not have complex multiplication.Modular form 123840.2.a.bh
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.