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SageMath
E = EllipticCurve("bm1")
E.isogeny_class()
Elliptic curves in class 190608bm
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 190608.j5 | 190608bm1 | \([0, -1, 0, -860744, 306371568]\) | \(365986170577/1765632\) | \(340237160291500032\) | \([2]\) | \(2211840\) | \(2.2127\) | \(\Gamma_0(N)\)-optimal |
| 190608.j4 | 190608bm2 | \([0, -1, 0, -1322824, -58117136]\) | \(1328460616657/761097744\) | \(146663480908154732544\) | \([2, 2]\) | \(4423680\) | \(2.5592\) | |
| 190608.j6 | 190608bm3 | \([0, -1, 0, 5261816, -468998672]\) | \(83608233481583/48873824868\) | \(-9417982561299532627968\) | \([2]\) | \(8847360\) | \(2.9058\) | |
| 190608.j2 | 190608bm4 | \([0, -1, 0, -15300744, -22981905936]\) | \(2055795133410577/5109104484\) | \(984524069154121334784\) | \([2, 2]\) | \(8847360\) | \(2.9058\) | |
| 190608.j3 | 190608bm5 | \([0, -1, 0, -9582504, -40374504720]\) | \(-504985875929137/3362745482118\) | \(-648000814223405285203968\) | \([2]\) | \(17694720\) | \(3.2524\) | |
| 190608.j1 | 190608bm6 | \([0, -1, 0, -244665704, -1472935437072]\) | \(8405459297332260337/52107462\) | \(10041104205676634112\) | \([2]\) | \(17694720\) | \(3.2524\) |
Rank
sage: E.rank()
The elliptic curves in class 190608bm have rank \(1\).
Complex multiplication
The elliptic curves in class 190608bm do not have complex multiplication.Modular form 190608.2.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.
