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SageMath
E = EllipticCurve("d1")
E.isogeny_class()
Elliptic curves in class 3450d
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
3450.d5 | 3450d1 | \([1, 1, 0, -10500, 450000]\) | \(-8194759433281/965779200\) | \(-15090300000000\) | \([2]\) | \(9216\) | \(1.2646\) | \(\Gamma_0(N)\)-optimal |
3450.d4 | 3450d2 | \([1, 1, 0, -172500, 27504000]\) | \(36330796409313601/428490000\) | \(6695156250000\) | \([2, 2]\) | \(18432\) | \(1.6112\) | |
3450.d3 | 3450d3 | \([1, 1, 0, -177000, 25987500]\) | \(39248884582600321/3935264062500\) | \(61488500976562500\) | \([2, 2]\) | \(36864\) | \(1.9577\) | |
3450.d1 | 3450d4 | \([1, 1, 0, -2760000, 1763716500]\) | \(148809678420065817601/20700\) | \(323437500\) | \([2]\) | \(36864\) | \(1.9577\) | |
3450.d2 | 3450d5 | \([1, 1, 0, -645750, -171356250]\) | \(1905890658841300321/293666194803750\) | \(4588534293808593750\) | \([2]\) | \(73728\) | \(2.3043\) | |
3450.d6 | 3450d6 | \([1, 1, 0, 219750, 126365250]\) | \(75108181893694559/484313964843750\) | \(-7567405700683593750\) | \([2]\) | \(73728\) | \(2.3043\) |
Rank
sage: E.rank()
The elliptic curves in class 3450d have rank \(0\).
Complex multiplication
The elliptic curves in class 3450d do not have complex multiplication.Modular form 3450.2.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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 2 & 2 \\ 4 & 2 & 4 & 1 & 8 & 8 \\ 8 & 4 & 2 & 8 & 1 & 4 \\ 8 & 4 & 2 & 8 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.