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SageMath
E = EllipticCurve("bi1")
E.isogeny_class()
Elliptic curves in class 187425bi
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
187425.bn4 | 187425bi1 | \([1, -1, 1, -1995755, -2465701878]\) | \(-656008386769/1581036975\) | \(-2118740757864437109375\) | \([2]\) | \(7077888\) | \(2.7802\) | \(\Gamma_0(N)\)-optimal |
187425.bn3 | 187425bi2 | \([1, -1, 1, -42181880, -105342181878]\) | \(6193921595708449/6452105625\) | \(8646438620914541015625\) | \([2, 2]\) | \(14155776\) | \(3.1268\) | |
187425.bn2 | 187425bi3 | \([1, -1, 1, -52600505, -49310816628]\) | \(12010404962647729/6166198828125\) | \(8263296168797296142578125\) | \([2]\) | \(28311552\) | \(3.4734\) | |
187425.bn1 | 187425bi4 | \([1, -1, 1, -674741255, -6745950500628]\) | \(25351269426118370449/27551475\) | \(36921611540288671875\) | \([2]\) | \(28311552\) | \(3.4734\) |
Rank
sage: E.rank()
The elliptic curves in class 187425bi have rank \(0\).
Complex multiplication
The elliptic curves in class 187425bi do not have complex multiplication.Modular form 187425.2.a.bi
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.