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SageMath
E = EllipticCurve("f1")
E.isogeny_class()
Elliptic curves in class 50430.f
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
50430.f1 | 50430f7 | \([1, 1, 0, -8965648, -10336598792]\) | \(16778985534208729/81000\) | \(384758443521000\) | \([2]\) | \(1658880\) | \(2.4210\) | |
50430.f2 | 50430f8 | \([1, 1, 0, -762368, -35188728]\) | \(10316097499609/5859375000\) | \(27832642037109375000\) | \([2]\) | \(1658880\) | \(2.4210\) | |
50430.f3 | 50430f6 | \([1, 1, 0, -560648, -161505792]\) | \(4102915888729/9000000\) | \(42750938169000000\) | \([2, 2]\) | \(829440\) | \(2.0744\) | |
50430.f4 | 50430f5 | \([1, 1, 0, -485003, 129803103]\) | \(2656166199049/33750\) | \(160316018133750\) | \([2]\) | \(552960\) | \(1.8716\) | |
50430.f5 | 50430f4 | \([1, 1, 0, -115183, -13007933]\) | \(35578826569/5314410\) | \(25244001479412810\) | \([2]\) | \(552960\) | \(1.8716\) | |
50430.f6 | 50430f2 | \([1, 1, 0, -31133, 1902537]\) | \(702595369/72900\) | \(346282599168900\) | \([2, 2]\) | \(276480\) | \(1.5251\) | |
50430.f7 | 50430f3 | \([1, 1, 0, -22728, -4325568]\) | \(-273359449/1536000\) | \(-7296160114176000\) | \([2]\) | \(414720\) | \(1.7278\) | |
50430.f8 | 50430f1 | \([1, 1, 0, 2487, 147573]\) | \(357911/2160\) | \(-10260225160560\) | \([2]\) | \(138240\) | \(1.1785\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 50430.f have rank \(1\).
Complex multiplication
The elliptic curves in class 50430.f do not have complex multiplication.Modular form 50430.2.a.f
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.