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SageMath
E = EllipticCurve("d1")
E.isogeny_class()
Elliptic curves in class 2850d
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 2850.c3 | 2850d1 | \([1, 1, 0, -646775, 201775125]\) | \(-1914980734749238129/20440940544000\) | \(-319389696000000000\) | \([2]\) | \(69120\) | \(2.1752\) | \(\Gamma_0(N)\)-optimal |
| 2850.c2 | 2850d2 | \([1, 1, 0, -10374775, 12857903125]\) | \(7903870428425797297009/886464000000\) | \(13851000000000000\) | \([2]\) | \(138240\) | \(2.5218\) | |
| 2850.c4 | 2850d3 | \([1, 1, 0, 2137225, 1052623125]\) | \(69096190760262356111/70568821500000000\) | \(-1102637835937500000000\) | \([2]\) | \(207360\) | \(2.7245\) | |
| 2850.c1 | 2850d4 | \([1, 1, 0, -11580775, 9681245125]\) | \(10993009831928446009969/3767761230468750000\) | \(58871269226074218750000\) | \([2]\) | \(414720\) | \(3.0711\) |
Rank
sage: E.rank()
The elliptic curves in class 2850d have rank \(0\).
Complex multiplication
The elliptic curves in class 2850d do not have complex multiplication.Modular form 2850.2.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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
