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SageMath
E = EllipticCurve("go1")
E.isogeny_class()
Elliptic curves in class 493680.go
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
493680.go1 | 493680go6 | \([0, 1, 0, -565583080, -5176745911372]\) | \(2757381641970898311361/379829992662450\) | \(2756165638680914265907200\) | \([2]\) | \(141557760\) | \(3.7073\) | |
493680.go2 | 493680go4 | \([0, 1, 0, -38507080, -65584524172]\) | \(870220733067747361/247623269602500\) | \(1796832162284644362240000\) | \([2, 2]\) | \(70778880\) | \(3.3607\) | |
493680.go3 | 493680go2 | \([0, 1, 0, -14307080, 20015715828]\) | \(44633474953947361/1967006250000\) | \(14273214706713600000000\) | \([2, 2]\) | \(35389440\) | \(3.0141\) | |
493680.go4 | 493680go1 | \([0, 1, 0, -14152200, 20487232500]\) | \(43199583152847841/89760000\) | \(651326731714560000\) | \([2]\) | \(17694720\) | \(2.6675\) | \(\Gamma_0(N)\)-optimal* |
493680.go5 | 493680go3 | \([0, 1, 0, 7414840, 75441366900]\) | \(6213165856218719/342407226562500\) | \(-2484614302500000000000000\) | \([2]\) | \(70778880\) | \(3.3607\) | |
493680.go6 | 493680go5 | \([0, 1, 0, 101368920, -432451296972]\) | \(15875306080318016639/20322604533582450\) | \(-147467196867042749253427200\) | \([2]\) | \(141557760\) | \(3.7073\) |
Rank
sage: E.rank()
The elliptic curves in class 493680.go have rank \(1\).
Complex multiplication
The elliptic curves in class 493680.go do not have complex multiplication.Modular form 493680.2.a.go
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.