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SageMath
E = EllipticCurve("bk1")
E.isogeny_class()
Elliptic curves in class 40656bk
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 40656.l6 | 40656bk1 | \([0, -1, 0, 1896, -7632]\) | \(103823/63\) | \(-457147772928\) | \([2]\) | \(40960\) | \(0.92658\) | \(\Gamma_0(N)\)-optimal |
| 40656.l5 | 40656bk2 | \([0, -1, 0, -7784, -54096]\) | \(7189057/3969\) | \(28800309694464\) | \([2, 2]\) | \(81920\) | \(1.2732\) | |
| 40656.l3 | 40656bk3 | \([0, -1, 0, -75544, 7968688]\) | \(6570725617/45927\) | \(333260726464512\) | \([2]\) | \(163840\) | \(1.6197\) | |
| 40656.l2 | 40656bk4 | \([0, -1, 0, -94904, -11205456]\) | \(13027640977/21609\) | \(156801686114304\) | \([2, 2]\) | \(163840\) | \(1.6197\) | |
| 40656.l4 | 40656bk5 | \([0, -1, 0, -65864, -18221520]\) | \(-4354703137/17294403\) | \(-125493616120147968\) | \([2]\) | \(327680\) | \(1.9663\) | |
| 40656.l1 | 40656bk6 | \([0, -1, 0, -1517864, -719270352]\) | \(53297461115137/147\) | \(1066678136832\) | \([2]\) | \(327680\) | \(1.9663\) |
Rank
sage: E.rank()
The elliptic curves in class 40656bk have rank \(1\).
Complex multiplication
The elliptic curves in class 40656bk do not have complex multiplication.Modular form 40656.2.a.bk
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 8 & 8 \\ 4 & 2 & 4 & 1 & 2 & 2 \\ 8 & 4 & 8 & 2 & 1 & 4 \\ 8 & 4 & 8 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
