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SageMath
E = EllipticCurve("ef1")
E.isogeny_class()
Elliptic curves in class 54450.ef
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
54450.ef1 | 54450gf7 | \([1, -1, 1, -145205105, -673437860103]\) | \(16778985534208729/81000\) | \(1634514148265625000\) | \([2]\) | \(6635520\) | \(3.1171\) | |
54450.ef2 | 54450gf8 | \([1, -1, 1, -12347105, -2269736103]\) | \(10316097499609/5859375000\) | \(118237423919677734375000\) | \([2]\) | \(6635520\) | \(3.1171\) | |
54450.ef3 | 54450gf6 | \([1, -1, 1, -9080105, -10509110103]\) | \(4102915888729/9000000\) | \(181612683140625000000\) | \([2, 2]\) | \(3317760\) | \(2.7706\) | |
54450.ef4 | 54450gf5 | \([1, -1, 1, -7854980, 8475426897]\) | \(2656166199049/33750\) | \(681047561777343750\) | \([2]\) | \(2211840\) | \(2.5678\) | |
54450.ef5 | 54450gf4 | \([1, -1, 1, -1865480, -844235103]\) | \(35578826569/5314410\) | \(107240473267707656250\) | \([2]\) | \(2211840\) | \(2.5678\) | |
54450.ef6 | 54450gf2 | \([1, -1, 1, -504230, 124974897]\) | \(702595369/72900\) | \(1471062733439062500\) | \([2, 2]\) | \(1105920\) | \(2.2213\) | |
54450.ef7 | 54450gf3 | \([1, -1, 1, -368105, -281222103]\) | \(-273359449/1536000\) | \(-30995231256000000000\) | \([2]\) | \(1658880\) | \(2.4240\) | |
54450.ef8 | 54450gf1 | \([1, -1, 1, 40270, 9540897]\) | \(357911/2160\) | \(-43587043953750000\) | \([2]\) | \(552960\) | \(1.8747\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 54450.ef have rank \(0\).
Complex multiplication
The elliptic curves in class 54450.ef do not have complex multiplication.Modular form 54450.2.a.ef
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 4 & 2 & 12 & 3 & 6 & 4 & 12 \\ 4 & 1 & 2 & 3 & 12 & 6 & 4 & 12 \\ 2 & 2 & 1 & 6 & 6 & 3 & 2 & 6 \\ 12 & 3 & 6 & 1 & 4 & 2 & 12 & 4 \\ 3 & 12 & 6 & 4 & 1 & 2 & 12 & 4 \\ 6 & 6 & 3 & 2 & 2 & 1 & 6 & 2 \\ 4 & 4 & 2 & 12 & 12 & 6 & 1 & 3 \\ 12 & 12 & 6 & 4 & 4 & 2 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.