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SageMath
E = EllipticCurve("ce1")
E.isogeny_class()
Elliptic curves in class 106470ce
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
106470.cg7 | 106470ce1 | \([1, -1, 0, -756729, -253085715]\) | \(13619385906841/6048000\) | \(21281362266528000\) | \([2]\) | \(1769472\) | \(2.0916\) | \(\Gamma_0(N)\)-optimal |
106470.cg6 | 106470ce2 | \([1, -1, 0, -878409, -166133187]\) | \(21302308926361/8930250000\) | \(31423261471670250000\) | \([2, 2]\) | \(3538944\) | \(2.4382\) | |
106470.cg5 | 106470ce3 | \([1, -1, 0, -2239704, 981046080]\) | \(353108405631241/86318776320\) | \(303733655633154539520\) | \([2]\) | \(5308416\) | \(2.6409\) | |
106470.cg8 | 106470ce4 | \([1, -1, 0, 2924091, -1222467687]\) | \(785793873833639/637994920500\) | \(-2244940646059066000500\) | \([2]\) | \(7077888\) | \(2.7847\) | |
106470.cg4 | 106470ce5 | \([1, -1, 0, -6627789, 6453702945]\) | \(9150443179640281/184570312500\) | \(649455635575195312500\) | \([2]\) | \(7077888\) | \(2.7847\) | |
106470.cg2 | 106470ce6 | \([1, -1, 0, -33389784, 74264724288]\) | \(1169975873419524361/108425318400\) | \(381520912654438502400\) | \([2, 2]\) | \(10616832\) | \(2.9875\) | |
106470.cg3 | 106470ce7 | \([1, -1, 0, -30956184, 85547380608]\) | \(-932348627918877961/358766164249920\) | \(-1262406202112307244749120\) | \([2]\) | \(21233664\) | \(3.3340\) | |
106470.cg1 | 106470ce8 | \([1, -1, 0, -534224664, 4752763672320]\) | \(4791901410190533590281/41160000\) | \(144831493202760000\) | \([2]\) | \(21233664\) | \(3.3340\) |
Rank
sage: E.rank()
The elliptic curves in class 106470ce have rank \(1\).
Complex multiplication
The elliptic curves in class 106470ce do not have complex multiplication.Modular form 106470.2.a.ce
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 3 & 4 & 4 & 6 & 12 & 12 \\ 2 & 1 & 6 & 2 & 2 & 3 & 6 & 6 \\ 3 & 6 & 1 & 12 & 12 & 2 & 4 & 4 \\ 4 & 2 & 12 & 1 & 4 & 6 & 3 & 12 \\ 4 & 2 & 12 & 4 & 1 & 6 & 12 & 3 \\ 6 & 3 & 2 & 6 & 6 & 1 & 2 & 2 \\ 12 & 6 & 4 & 3 & 12 & 2 & 1 & 4 \\ 12 & 6 & 4 & 12 & 3 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.