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SageMath
E = EllipticCurve("fu1")
E.isogeny_class()
Elliptic curves in class 84150fu
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 84150.fo4 | 84150fu1 | \([1, -1, 1, -1644755, -811482253]\) | \(43199583152847841/89760000\) | \(1022422500000000\) | \([2]\) | \(1179648\) | \(2.1295\) | \(\Gamma_0(N)\)-optimal |
| 84150.fo3 | 84150fu2 | \([1, -1, 1, -1662755, -792798253]\) | \(44633474953947361/1967006250000\) | \(22405430566406250000\) | \([2, 2]\) | \(2359296\) | \(2.4760\) | |
| 84150.fo5 | 84150fu3 | \([1, -1, 1, 861745, -2989113253]\) | \(6213165856218719/342407226562500\) | \(-3900232315063476562500\) | \([2]\) | \(4718592\) | \(2.8226\) | |
| 84150.fo2 | 84150fu4 | \([1, -1, 1, -4475255, 2599076747]\) | \(870220733067747361/247623269602500\) | \(2820583805315976562500\) | \([2, 2]\) | \(4718592\) | \(2.8226\) | |
| 84150.fo6 | 84150fu5 | \([1, -1, 1, 11780995, 17132164247]\) | \(15875306080318016639/20322604533582450\) | \(-231487167265337594531250\) | \([2]\) | \(9437184\) | \(3.1692\) | |
| 84150.fo1 | 84150fu6 | \([1, -1, 1, -65731505, 205112239247]\) | \(2757381641970898311361/379829992662450\) | \(4326501010170719531250\) | \([2]\) | \(9437184\) | \(3.1692\) |
Rank
sage: E.rank()
The elliptic curves in class 84150fu have rank \(1\).
Complex multiplication
The elliptic curves in class 84150fu do not have complex multiplication.Modular form 84150.2.a.fu
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.
