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SageMath
E = EllipticCurve("d1")
E.isogeny_class()
Elliptic curves in class 4410d
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 4410.p2 | 4410d1 | \([1, -1, 0, -5154, -140960]\) | \(4767078987/6860\) | \(21790947780\) | \([2]\) | \(4608\) | \(0.88616\) | \(\Gamma_0(N)\)-optimal |
| 4410.p3 | 4410d2 | \([1, -1, 0, -3684, -224162]\) | \(-1740992427/5882450\) | \(-18685737721350\) | \([2]\) | \(9216\) | \(1.2327\) | |
| 4410.p1 | 4410d3 | \([1, -1, 0, -20589, 1001573]\) | \(416832723/56000\) | \(129678374952000\) | \([2]\) | \(13824\) | \(1.4355\) | |
| 4410.p4 | 4410d4 | \([1, -1, 0, 32331, 5266925]\) | \(1613964717/6125000\) | \(-14183572260375000\) | \([2]\) | \(27648\) | \(1.7820\) |
Rank
sage: E.rank()
The elliptic curves in class 4410d have rank \(1\).
Complex multiplication
The elliptic curves in class 4410d do not have complex multiplication.Modular form 4410.2.a.d
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
