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SageMath
E = EllipticCurve("u1")
E.isogeny_class()
Elliptic curves in class 400400.u
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
400400.u1 | 400400u4 | \([0, 1, 0, -31594383408, 2160972222863188]\) | \(54497099771831721530744218729/16209843781074944000000\) | \(1037430001988796416000000000000\) | \([2]\) | \(836075520\) | \(4.7380\) | \(\Gamma_0(N)\)-optimal* |
400400.u2 | 400400u3 | \([0, 1, 0, -2234255408, 24318267791188]\) | \(19272683606216463573689449/7161126378530668544000\) | \(458312088225962786816000000000\) | \([2]\) | \(418037760\) | \(4.3914\) | \(\Gamma_0(N)\)-optimal* |
400400.u3 | 400400u2 | \([0, 1, 0, -1053257408, -9319955652812]\) | \(2019051077229077416165369/582160888682835862400\) | \(37258296875701495193600000000\) | \([2]\) | \(278691840\) | \(4.1886\) | \(\Gamma_0(N)\)-optimal* |
400400.u4 | 400400u1 | \([0, 1, 0, -965449408, -11545185988812]\) | \(1555006827939811751684089/221961497899581440\) | \(14205535865573212160000000\) | \([2]\) | \(139345920\) | \(3.8421\) | \(\Gamma_0(N)\)-optimal* |
Rank
sage: E.rank()
The elliptic curves in class 400400.u have rank \(1\).
Complex multiplication
The elliptic curves in class 400400.u do not have complex multiplication.Modular form 400400.2.a.u
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.