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SageMath
E = EllipticCurve("a1")
E.isogeny_class()
Elliptic curves in class 4140.a
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 4140.a1 | 4140e3 | \([0, 0, 0, -109488, 13944337]\) | \(12444451776495616/912525\) | \(10643691600\) | \([6]\) | \(12096\) | \(1.3748\) | |
| 4140.a2 | 4140e4 | \([0, 0, 0, -109263, 14004502]\) | \(-772993034343376/6661615005\) | \(-1243217238693120\) | \([6]\) | \(24192\) | \(1.7213\) | |
| 4140.a3 | 4140e1 | \([0, 0, 0, -1488, 15037]\) | \(31238127616/9703125\) | \(113177250000\) | \([2]\) | \(4032\) | \(0.82545\) | \(\Gamma_0(N)\)-optimal |
| 4140.a4 | 4140e2 | \([0, 0, 0, 4137, 101662]\) | \(41957807024/48205125\) | \(-8996233248000\) | \([2]\) | \(8064\) | \(1.1720\) |
Rank
sage: E.rank()
The elliptic curves in class 4140.a have rank \(0\).
Complex multiplication
The elliptic curves in class 4140.a do not have complex multiplication.Modular form 4140.2.a.a
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
