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SageMath
E = EllipticCurve("i1")
E.isogeny_class()
Elliptic curves in class 301665.i
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 301665.i1 | 301665i3 | \([1, 1, 1, -20643776, 36054594974]\) | \(201572375361968225161/250924004296875\) | \(1211162242256194921875\) | \([2]\) | \(18579456\) | \(2.9546\) | |
| 301665.i2 | 301665i4 | \([1, 1, 1, -15095506, -22405257022]\) | \(78814590865112105641/706854753893925\) | \(3411852887787982235325\) | \([2]\) | \(18579456\) | \(2.9546\) | |
| 301665.i3 | 301665i2 | \([1, 1, 1, -1638881, 234168878]\) | \(100856960534879641/53429886680625\) | \(257895857899020875625\) | \([2, 2]\) | \(9289728\) | \(2.6081\) | |
| 301665.i4 | 301665i1 | \([1, 1, 1, 389964, 28849764]\) | \(1358742243975479/859964189175\) | \(-4150882887987592575\) | \([4]\) | \(4644864\) | \(2.2615\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 301665.i have rank \(1\).
Complex multiplication
The elliptic curves in class 301665.i do not have complex multiplication.Modular form 301665.2.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 LMFDB numbering.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
