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SageMath
E = EllipticCurve("f1")
E.isogeny_class()
Elliptic curves in class 3330f
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 3330.c5 | 3330f1 | \([1, -1, 0, -7605, -876875]\) | \(-66730743078481/419010969600\) | \(-305458996838400\) | \([2]\) | \(13824\) | \(1.4630\) | \(\Gamma_0(N)\)-optimal |
| 3330.c4 | 3330f2 | \([1, -1, 0, -191925, -32248139]\) | \(1072487167529950801/2554882560000\) | \(1862509386240000\) | \([2, 2]\) | \(27648\) | \(1.8096\) | |
| 3330.c1 | 3330f3 | \([1, -1, 0, -3069045, -2068673675]\) | \(4385367890843575421521/24975000000\) | \(18206775000000\) | \([2]\) | \(55296\) | \(2.1561\) | |
| 3330.c3 | 3330f4 | \([1, -1, 0, -263925, -5795339]\) | \(2788936974993502801/1593609593601600\) | \(1161741393735566400\) | \([2, 2]\) | \(55296\) | \(2.1561\) | |
| 3330.c2 | 3330f5 | \([1, -1, 0, -2728125, 1727522941]\) | \(3080272010107543650001/15465841417699560\) | \(11274598393502979240\) | \([2]\) | \(110592\) | \(2.5027\) | |
| 3330.c6 | 3330f6 | \([1, -1, 0, 1048275, -46998419]\) | \(174751791402194852399/102423900876336360\) | \(-74667023738849206440\) | \([2]\) | \(110592\) | \(2.5027\) |
Rank
sage: E.rank()
The elliptic curves in class 3330f have rank \(1\).
Complex multiplication
The elliptic curves in class 3330f do not have complex multiplication.Modular form 3330.2.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 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.
