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SageMath
E = EllipticCurve("d1")
E.isogeny_class()
Elliptic curves in class 9405.d
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 9405.d1 | 9405i3 | \([1, -1, 1, -9151103, 9375314212]\) | \(116256292809537371612841/15216540068579856875\) | \(11092857709994715661875\) | \([2]\) | \(565248\) | \(2.9583\) | |
| 9405.d2 | 9405i2 | \([1, -1, 1, -8841728, 10121402962]\) | \(104859453317683374662841/2223652969140625\) | \(1621043014503515625\) | \([2, 2]\) | \(282624\) | \(2.6117\) | |
| 9405.d3 | 9405i1 | \([1, -1, 1, -8841683, 10121511106]\) | \(104857852278310619039721/47155625\) | \(34376450625\) | \([2]\) | \(141312\) | \(2.2651\) | \(\Gamma_0(N)\)-optimal |
| 9405.d4 | 9405i4 | \([1, -1, 1, -8533073, 10860569956]\) | \(-94256762600623910012361/15323275604248046875\) | \(-11170667915496826171875\) | \([2]\) | \(565248\) | \(2.9583\) |
Rank
sage: E.rank()
The elliptic curves in class 9405.d have rank \(0\).
Complex multiplication
The elliptic curves in class 9405.d do not have complex multiplication.Modular form 9405.2.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.
