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SageMath
E = EllipticCurve("q1")
E.isogeny_class()
Elliptic curves in class 2070q
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
2070.r3 | 2070q1 | \([1, -1, 1, -5432, 154811]\) | \(24310870577209/114462720\) | \(83443322880\) | \([4]\) | \(3840\) | \(0.94530\) | \(\Gamma_0(N)\)-optimal |
2070.r2 | 2070q2 | \([1, -1, 1, -8312, -24901]\) | \(87109155423289/49979073600\) | \(36434744654400\) | \([2, 2]\) | \(7680\) | \(1.2919\) | |
2070.r1 | 2070q3 | \([1, -1, 1, -95792, -11362309]\) | \(133345896593725369/340006815000\) | \(247864968135000\) | \([2]\) | \(15360\) | \(1.6384\) | |
2070.r4 | 2070q4 | \([1, -1, 1, 33088, -223621]\) | \(5495662324535111/3207841648920\) | \(-2338516562062680\) | \([2]\) | \(15360\) | \(1.6384\) |
Rank
sage: E.rank()
The elliptic curves in class 2070q have rank \(0\).
Complex multiplication
The elliptic curves in class 2070q do not have complex multiplication.Modular form 2070.2.a.q
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 Cremona numbering.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.