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SageMath
E = EllipticCurve("d1")
E.isogeny_class()
Elliptic curves in class 1815.d
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
1815.d1 | 1815a7 | \([1, 1, 0, -261362, 51320691]\) | \(1114544804970241/405\) | \(717482205\) | \([2]\) | \(5120\) | \(1.4898\) | |
1815.d2 | 1815a5 | \([1, 1, 0, -16337, 796536]\) | \(272223782641/164025\) | \(290580293025\) | \([2, 2]\) | \(2560\) | \(1.1432\) | |
1815.d3 | 1815a8 | \([1, 1, 0, -13312, 1104481]\) | \(-147281603041/215233605\) | \(-381299460507405\) | \([2]\) | \(5120\) | \(1.4898\) | |
1815.d4 | 1815a3 | \([1, 1, 0, -9682, -370751]\) | \(56667352321/15\) | \(26573415\) | \([2]\) | \(1280\) | \(0.79667\) | |
1815.d5 | 1815a4 | \([1, 1, 0, -1212, 7011]\) | \(111284641/50625\) | \(89685275625\) | \([2, 2]\) | \(1280\) | \(0.79667\) | |
1815.d6 | 1815a2 | \([1, 1, 0, -607, -5936]\) | \(13997521/225\) | \(398601225\) | \([2, 2]\) | \(640\) | \(0.45010\) | |
1815.d7 | 1815a1 | \([1, 1, 0, -2, -249]\) | \(-1/15\) | \(-26573415\) | \([2]\) | \(320\) | \(0.10352\) | \(\Gamma_0(N)\)-optimal |
1815.d8 | 1815a6 | \([1, 1, 0, 4233, 58194]\) | \(4733169839/3515625\) | \(-6228144140625\) | \([2]\) | \(2560\) | \(1.1432\) |
Rank
sage: E.rank()
The elliptic curves in class 1815.d have rank \(1\).
Complex multiplication
The elliptic curves in class 1815.d do not have complex multiplication.Modular form 1815.2.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.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 16 & 4 & 8 & 16 & 8 \\ 2 & 1 & 2 & 8 & 2 & 4 & 8 & 4 \\ 4 & 2 & 1 & 16 & 4 & 8 & 16 & 8 \\ 16 & 8 & 16 & 1 & 4 & 2 & 4 & 8 \\ 4 & 2 & 4 & 4 & 1 & 2 & 4 & 2 \\ 8 & 4 & 8 & 2 & 2 & 1 & 2 & 4 \\ 16 & 8 & 16 & 4 & 4 & 2 & 1 & 8 \\ 8 & 4 & 8 & 8 & 2 & 4 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.