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SageMath
E = EllipticCurve("a1")
E.isogeny_class()
Elliptic curves in class 10010.a
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
10010.a1 | 10010f4 | \([1, 0, 1, -78985959, -270129426454]\) | \(54497099771831721530744218729/16209843781074944000000\) | \(16209843781074944000000\) | \([2]\) | \(1451520\) | \(3.2401\) | |
10010.a2 | 10010f3 | \([1, 0, 1, -5585639, -3040342038]\) | \(19272683606216463573689449/7161126378530668544000\) | \(7161126378530668544000\) | \([2]\) | \(725760\) | \(2.8935\) | |
10010.a3 | 10010f2 | \([1, 0, 1, -2633144, 1164731142]\) | \(2019051077229077416165369/582160888682835862400\) | \(582160888682835862400\) | \([6]\) | \(483840\) | \(2.6908\) | |
10010.a4 | 10010f1 | \([1, 0, 1, -2413624, 1442906886]\) | \(1555006827939811751684089/221961497899581440\) | \(221961497899581440\) | \([6]\) | \(241920\) | \(2.3442\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 10010.a have rank \(0\).
Complex multiplication
The elliptic curves in class 10010.a do not have complex multiplication.Modular form 10010.2.a.a
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.