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SageMath
E = EllipticCurve("d1")
E.isogeny_class()
Elliptic curves in class 47040.d
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 47040.d1 | 47040eq4 | \([0, -1, 0, -4370081, 2178798945]\) | \(2394165105226952/854262178245\) | \(3293285286161481891840\) | \([2]\) | \(2949120\) | \(2.8298\) | |
| 47040.d2 | 47040eq2 | \([0, -1, 0, -3889881, 2953553625]\) | \(13507798771700416/3544416225\) | \(1708019812167782400\) | \([2, 2]\) | \(1474560\) | \(2.4832\) | |
| 47040.d3 | 47040eq1 | \([0, -1, 0, -3889636, 2953944106]\) | \(864335783029582144/59535\) | \(448270925760\) | \([2]\) | \(737280\) | \(2.1366\) | \(\Gamma_0(N)\)-optimal |
| 47040.d4 | 47040eq3 | \([0, -1, 0, -3413601, 3703313601]\) | \(-1141100604753992/875529151875\) | \(-3375272073263247360000\) | \([2]\) | \(2949120\) | \(2.8298\) |
Rank
sage: E.rank()
The elliptic curves in class 47040.d have rank \(1\).
Complex multiplication
The elliptic curves in class 47040.d do not have complex multiplication.Modular form 47040.2.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 LMFDB 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 LMFDB labels.
