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SageMath
E = EllipticCurve("s1")
E.isogeny_class()
Elliptic curves in class 3630s
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 3630.s3 | 3630s1 | \([1, 1, 1, -2725, 45035]\) | \(1263214441/211200\) | \(374153683200\) | \([4]\) | \(7680\) | \(0.94179\) | \(\Gamma_0(N)\)-optimal |
| 3630.s2 | 3630s2 | \([1, 1, 1, -12405, -493173]\) | \(119168121961/10890000\) | \(19292299290000\) | \([2, 2]\) | \(15360\) | \(1.2884\) | |
| 3630.s1 | 3630s3 | \([1, 1, 1, -193905, -32945373]\) | \(455129268177961/4392300\) | \(7781227380300\) | \([2]\) | \(30720\) | \(1.6349\) | |
| 3630.s4 | 3630s4 | \([1, 1, 1, 14215, -2292685]\) | \(179310732119/1392187500\) | \(-2466345079687500\) | \([2]\) | \(30720\) | \(1.6349\) |
Rank
sage: E.rank()
The elliptic curves in class 3630s have rank \(0\).
Complex multiplication
The elliptic curves in class 3630s do not have complex multiplication.Modular form 3630.2.a.s
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.
