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SageMath
E = EllipticCurve("d1")
E.isogeny_class()
Elliptic curves in class 1470d
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
1470.d8 | 1470d1 | \([1, 1, 0, 73, -699]\) | \(357911/2160\) | \(-254121840\) | \([2]\) | \(576\) | \(0.29467\) | \(\Gamma_0(N)\)-optimal |
1470.d6 | 1470d2 | \([1, 1, 0, -907, -9911]\) | \(702595369/72900\) | \(8576612100\) | \([2, 2]\) | \(1152\) | \(0.64124\) | |
1470.d7 | 1470d3 | \([1, 1, 0, -662, 21204]\) | \(-273359449/1536000\) | \(-180708864000\) | \([2]\) | \(1728\) | \(0.84397\) | |
1470.d4 | 1470d4 | \([1, 1, 0, -14137, -652889]\) | \(2656166199049/33750\) | \(3970653750\) | \([2]\) | \(2304\) | \(0.98781\) | |
1470.d5 | 1470d5 | \([1, 1, 0, -3357, 63099]\) | \(35578826569/5314410\) | \(625235022090\) | \([2]\) | \(2304\) | \(0.98781\) | |
1470.d3 | 1470d6 | \([1, 1, 0, -16342, 795796]\) | \(4102915888729/9000000\) | \(1058841000000\) | \([2, 2]\) | \(3456\) | \(1.1905\) | |
1470.d2 | 1470d7 | \([1, 1, 0, -22222, 164284]\) | \(10316097499609/5859375000\) | \(689349609375000\) | \([2]\) | \(6912\) | \(1.5371\) | |
1470.d1 | 1470d8 | \([1, 1, 0, -261342, 51314796]\) | \(16778985534208729/81000\) | \(9529569000\) | \([2]\) | \(6912\) | \(1.5371\) |
Rank
sage: E.rank()
The elliptic curves in class 1470d have rank \(1\).
Complex multiplication
The elliptic curves in class 1470d do not have complex multiplication.Modular form 1470.2.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.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 3 & 4 & 4 & 6 & 12 & 12 \\ 2 & 1 & 6 & 2 & 2 & 3 & 6 & 6 \\ 3 & 6 & 1 & 12 & 12 & 2 & 4 & 4 \\ 4 & 2 & 12 & 1 & 4 & 6 & 3 & 12 \\ 4 & 2 & 12 & 4 & 1 & 6 & 12 & 3 \\ 6 & 3 & 2 & 6 & 6 & 1 & 2 & 2 \\ 12 & 6 & 4 & 3 & 12 & 2 & 1 & 4 \\ 12 & 6 & 4 & 12 & 3 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.