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SageMath
E = EllipticCurve("bm1")
E.isogeny_class()
Elliptic curves in class 5850bm
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 5850.bo6 | 5850bm1 | \([1, -1, 1, 3370, -31003]\) | \(371694959/249600\) | \(-2843100000000\) | \([2]\) | \(12288\) | \(1.0784\) | \(\Gamma_0(N)\)-optimal |
| 5850.bo5 | 5850bm2 | \([1, -1, 1, -14630, -247003]\) | \(30400540561/15210000\) | \(173251406250000\) | \([2, 2]\) | \(24576\) | \(1.4250\) | |
| 5850.bo2 | 5850bm3 | \([1, -1, 1, -190130, -31837003]\) | \(66730743078481/60937500\) | \(694116210937500\) | \([2]\) | \(49152\) | \(1.7715\) | |
| 5850.bo3 | 5850bm4 | \([1, -1, 1, -127130, 17302997]\) | \(19948814692561/231344100\) | \(2635153889062500\) | \([2, 2]\) | \(49152\) | \(1.7715\) | |
| 5850.bo1 | 5850bm5 | \([1, -1, 1, -2028380, 1112422997]\) | \(81025909800741361/11088090\) | \(126300275156250\) | \([2]\) | \(98304\) | \(2.1181\) | |
| 5850.bo4 | 5850bm6 | \([1, -1, 1, -25880, 44032997]\) | \(-168288035761/73415764890\) | \(-836251446950156250\) | \([2]\) | \(98304\) | \(2.1181\) |
Rank
sage: E.rank()
The elliptic curves in class 5850bm have rank \(1\).
Complex multiplication
The elliptic curves in class 5850bm do not have complex multiplication.Modular form 5850.2.a.bm
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.
