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SageMath
E = EllipticCurve("bf1")
E.isogeny_class()
Elliptic curves in class 185130bf
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 185130.dp4 | 185130bf1 | \([1, -1, 1, -7960613, 8647031517]\) | \(43199583152847841/89760000\) | \(115922164897440000\) | \([2]\) | \(5898240\) | \(2.5237\) | \(\Gamma_0(N)\)-optimal |
| 185130.dp3 | 185130bf2 | \([1, -1, 1, -8047733, 8448153981]\) | \(44633474953947361/1967006250000\) | \(2540325566697806250000\) | \([2, 2]\) | \(11796480\) | \(2.8703\) | |
| 185130.dp5 | 185130bf3 | \([1, -1, 1, 4170847, 31824741237]\) | \(6213165856218719/342407226562500\) | \(-442207965459594726562500\) | \([2]\) | \(23592960\) | \(3.2168\) | |
| 185130.dp2 | 185130bf4 | \([1, -1, 1, -21660233, -27657641019]\) | \(870220733067747361/247623269602500\) | \(319797521070680112322500\) | \([2, 2]\) | \(23592960\) | \(3.2168\) | |
| 185130.dp6 | 185130bf5 | \([1, -1, 1, 57020017, -182468900919]\) | \(15875306080318016639/20322604533582450\) | \(-26245992801775918995544050\) | \([2]\) | \(47185920\) | \(3.5634\) | |
| 185130.dp1 | 185130bf6 | \([1, -1, 1, -318140483, -2183780611119]\) | \(2757381641970898311361/379829992662450\) | \(490538269189059204064050\) | \([2]\) | \(47185920\) | \(3.5634\) |
Rank
sage: E.rank()
The elliptic curves in class 185130bf have rank \(1\).
Complex multiplication
The elliptic curves in class 185130bf do not have complex multiplication.Modular form 185130.2.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 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.
