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SageMath
E = EllipticCurve("be1")
E.isogeny_class()
Elliptic curves in class 134640.be
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 134640.be1 | 134640cd6 | \([0, 0, 0, -42068163, -105009052862]\) | \(2757381641970898311361/379829992662450\) | \(1134166280810193100800\) | \([2]\) | \(9437184\) | \(3.0576\) | |
| 134640.be2 | 134640cd4 | \([0, 0, 0, -2864163, -1330154462]\) | \(870220733067747361/247623269602500\) | \(739399121060751360000\) | \([2, 2]\) | \(4718592\) | \(2.7110\) | |
| 134640.be3 | 134640cd2 | \([0, 0, 0, -1064163, 406125538]\) | \(44633474953947361/1967006250000\) | \(5873449190400000000\) | \([2, 2]\) | \(2359296\) | \(2.3645\) | |
| 134640.be4 | 134640cd1 | \([0, 0, 0, -1052643, 415689442]\) | \(43199583152847841/89760000\) | \(268021923840000\) | \([2]\) | \(1179648\) | \(2.0179\) | \(\Gamma_0(N)\)-optimal |
| 134640.be5 | 134640cd3 | \([0, 0, 0, 551517, 1530315682]\) | \(6213165856218719/342407226562500\) | \(-1022422500000000000000\) | \([2]\) | \(4718592\) | \(2.7110\) | |
| 134640.be6 | 134640cd5 | \([0, 0, 0, 7539837, -8773176062]\) | \(15875306080318016639/20322604533582450\) | \(-60682971975604658380800\) | \([2]\) | \(9437184\) | \(3.0576\) |
Rank
sage: E.rank()
The elliptic curves in class 134640.be have rank \(1\).
Complex multiplication
The elliptic curves in class 134640.be do not have complex multiplication.Modular form 134640.2.a.be
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 LMFDB numbering.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 8 & 8 & 4 \\ 2 & 1 & 2 & 4 & 4 & 2 \\ 4 & 2 & 1 & 2 & 2 & 4 \\ 8 & 4 & 2 & 1 & 4 & 8 \\ 8 & 4 & 2 & 4 & 1 & 8 \\ 4 & 2 & 4 & 8 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
