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SageMath
E = EllipticCurve("p1")
E.isogeny_class()
Elliptic curves in class 201810.p
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
201810.p1 | 201810cg7 | \([1, 1, 0, -337534452, 2386710180624]\) | \(4791901410190533590281/41160000\) | \(36529651509960000\) | \([2]\) | \(34836480\) | \(3.2193\) | |
201810.p2 | 201810cg6 | \([1, 1, 0, -21096372, 37284011856]\) | \(1169975873419524361/108425318400\) | \(96227869193597030400\) | \([2, 2]\) | \(17418240\) | \(2.8727\) | |
201810.p3 | 201810cg8 | \([1, 1, 0, -19558772, 42951297936]\) | \(-932348627918877961/358766164249920\) | \(-318406291390054603955520\) | \([2]\) | \(34836480\) | \(3.2193\) | |
201810.p4 | 201810cg4 | \([1, 1, 0, -4187577, 3238581249]\) | \(9150443179640281/184570312500\) | \(163806831747070312500\) | \([2]\) | \(11612160\) | \(2.6700\) | |
201810.p5 | 201810cg3 | \([1, 1, 0, -1415092, 491827024]\) | \(353108405631241/86318776320\) | \(76608231723415633920\) | \([2]\) | \(8709120\) | \(2.5261\) | |
201810.p6 | 201810cg2 | \([1, 1, 0, -554997, -83776419]\) | \(21302308926361/8930250000\) | \(7925629747250250000\) | \([2, 2]\) | \(5806080\) | \(2.3234\) | |
201810.p7 | 201810cg1 | \([1, 1, 0, -478117, -127398131]\) | \(13619385906841/6048000\) | \(5367622262688000\) | \([2]\) | \(2903040\) | \(1.9768\) | \(\Gamma_0(N)\)-optimal |
201810.p8 | 201810cg5 | \([1, 1, 0, 1847503, -612806919]\) | \(785793873833639/637994920500\) | \(-566222840403052360500\) | \([2]\) | \(11612160\) | \(2.6700\) |
Rank
sage: E.rank()
The elliptic curves in class 201810.p have rank \(0\).
Complex multiplication
The elliptic curves in class 201810.p do not have complex multiplication.Modular form 201810.2.a.p
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 3 & 4 & 6 & 12 & 12 \\ 2 & 1 & 2 & 6 & 2 & 3 & 6 & 6 \\ 4 & 2 & 1 & 12 & 4 & 6 & 12 & 3 \\ 3 & 6 & 12 & 1 & 12 & 2 & 4 & 4 \\ 4 & 2 & 4 & 12 & 1 & 6 & 3 & 12 \\ 6 & 3 & 6 & 2 & 6 & 1 & 2 & 2 \\ 12 & 6 & 12 & 4 & 3 & 2 & 1 & 4 \\ 12 & 6 & 3 & 4 & 12 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.