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SageMath
E = EllipticCurve("i1")
E.isogeny_class()
Elliptic curves in class 218405i
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 218405.j3 | 218405i1 | \([1, -1, 0, -42912615719, -3421560575872400]\) | \(104857852278310619039721/47155625\) | \(3930168966384808255625\) | \([2]\) | \(190771200\) | \(4.3870\) | \(\Gamma_0(N)\)-optimal |
| 218405.j2 | 218405i2 | \([1, -1, 0, -42912834124, -3421524006008157]\) | \(104859453317683374662841/2223652969140625\) | \(185329573965479623799156640625\) | \([2, 2]\) | \(381542400\) | \(4.7336\) | |
| 218405.j1 | 218405i3 | \([1, -1, 0, -44414368499, -3169227491421282]\) | \(116256292809537371612841/15216540068579856875\) | \(1268217175645185816607984176756875\) | \([2]\) | \(763084800\) | \(5.0801\) | |
| 218405.j4 | 218405i4 | \([1, -1, 0, -41414794229, -3671480051904340]\) | \(-94256762600623910012361/15323275604248046875\) | \(-1277113011293500924862518310546875\) | \([2]\) | \(763084800\) | \(5.0801\) |
Rank
sage: E.rank()
The elliptic curves in class 218405i have rank \(0\).
Complex multiplication
The elliptic curves in class 218405i do not have complex multiplication.Modular form 218405.2.a.i
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.
