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SageMath
E = EllipticCurve("bm1")
E.isogeny_class()
Elliptic curves in class 35904bm
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 35904.cz5 | 35904bm1 | \([0, 1, 0, -18177, 708447]\) | \(2533811507137/625016832\) | \(163844412407808\) | \([2]\) | \(98304\) | \(1.4379\) | \(\Gamma_0(N)\)-optimal |
| 35904.cz4 | 35904bm2 | \([0, 1, 0, -100097, -11628705]\) | \(423108074414017/23284318464\) | \(6103844379426816\) | \([2, 2]\) | \(196608\) | \(1.7845\) | |
| 35904.cz6 | 35904bm3 | \([0, 1, 0, 68863, -46738593]\) | \(137763859017023/3683199928848\) | \(-965528762147930112\) | \([2]\) | \(393216\) | \(2.1311\) | |
| 35904.cz2 | 35904bm4 | \([0, 1, 0, -1579777, -764785825]\) | \(1663303207415737537/5483698704\) | \(1437518713061376\) | \([2, 2]\) | \(393216\) | \(2.1311\) | |
| 35904.cz3 | 35904bm5 | \([0, 1, 0, -1558017, -786854817]\) | \(-1595514095015181697/95635786040388\) | \(-25070347495771471872\) | \([2]\) | \(786432\) | \(2.4776\) | |
| 35904.cz1 | 35904bm6 | \([0, 1, 0, -25276417, -48921097633]\) | \(6812873765474836663297/74052\) | \(19412287488\) | \([2]\) | \(786432\) | \(2.4776\) |
Rank
sage: E.rank()
The elliptic curves in class 35904bm have rank \(0\).
Complex multiplication
The elliptic curves in class 35904bm do not have complex multiplication.Modular form 35904.2.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.
