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SageMath
E = EllipticCurve("bf1")
E.isogeny_class()
Elliptic curves in class 189618.bf
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 189618.bf1 | 189618t3 | \([1, 1, 1, -268085802, 1689390781983]\) | \(441453577446719855661097/4354701912\) | \(21019314381158808\) | \([2]\) | \(24772608\) | \(3.1638\) | |
| 189618.bf2 | 189618t2 | \([1, 1, 1, -16755762, 26390173311]\) | \(107784459654566688937/10704361149504\) | \(51667906735676252736\) | \([2, 2]\) | \(12386304\) | \(2.8172\) | |
| 189618.bf3 | 189618t4 | \([1, 1, 1, -15491642, 30541543391]\) | \(-85183593440646799657/34223681512621656\) | \(-165191173938255822775704\) | \([2]\) | \(24772608\) | \(3.1638\) | |
| 189618.bf4 | 189618t1 | \([1, 1, 1, -1126642, 345807743]\) | \(32765849647039657/8229948198912\) | \(39724388036042231808\) | \([2]\) | \(6193152\) | \(2.4707\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 189618.bf have rank \(0\).
Complex multiplication
The elliptic curves in class 189618.bf do not have complex multiplication.Modular form 189618.2.a.bf
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.
