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SageMath
E = EllipticCurve("ci1")
E.isogeny_class()
Elliptic curves in class 41616ci
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 41616.s5 | 41616ci1 | \([0, 0, 0, -1415811, 601361026]\) | \(4354703137/352512\) | \(25407089071333834752\) | \([2]\) | \(884736\) | \(2.4664\) | \(\Gamma_0(N)\)-optimal |
| 41616.s4 | 41616ci2 | \([0, 0, 0, -4745091, -3281245310]\) | \(163936758817/30338064\) | \(2186597603201668153344\) | \([2, 2]\) | \(1769472\) | \(2.8130\) | |
| 41616.s6 | 41616ci3 | \([0, 0, 0, 9404349, -19108808894]\) | \(1276229915423/2927177028\) | \(-210974512861855069913088\) | \([2]\) | \(3538944\) | \(3.1596\) | |
| 41616.s2 | 41616ci4 | \([0, 0, 0, -72163011, -235940487230]\) | \(576615941610337/27060804\) | \(1950391071991611408384\) | \([2, 2]\) | \(3538944\) | \(3.1596\) | |
| 41616.s3 | 41616ci5 | \([0, 0, 0, -68417571, -261524089694]\) | \(-491411892194497/125563633938\) | \(-9049922929100632024424448\) | \([2]\) | \(7077888\) | \(3.5061\) | |
| 41616.s1 | 41616ci6 | \([0, 0, 0, -1154595171, -15100548367646]\) | \(2361739090258884097/5202\) | \(374931001920724992\) | \([2]\) | \(7077888\) | \(3.5061\) |
Rank
sage: E.rank()
The elliptic curves in class 41616ci have rank \(1\).
Complex multiplication
The elliptic curves in class 41616ci do not have complex multiplication.Modular form 41616.2.a.ci
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 8 & 8 \\ 4 & 2 & 4 & 1 & 2 & 2 \\ 8 & 4 & 8 & 2 & 1 & 4 \\ 8 & 4 & 8 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
