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SageMath
E = EllipticCurve("kd1")
E.isogeny_class()
Elliptic curves in class 176400kd
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 176400.lf2 | 176400kd1 | \([0, 0, 0, -2061675, 1137988250]\) | \(4767078987/6860\) | \(1394620657920000000\) | \([2]\) | \(2654208\) | \(2.3840\) | \(\Gamma_0(N)\)-optimal |
| 176400.lf3 | 176400kd2 | \([0, 0, 0, -1473675, 1800664250]\) | \(-1740992427/5882450\) | \(-1195887214166400000000\) | \([2]\) | \(5308416\) | \(2.7306\) | |
| 176400.lf1 | 176400kd3 | \([0, 0, 0, -8235675, -7971405750]\) | \(416832723/56000\) | \(8299415996928000000000\) | \([2]\) | \(7962624\) | \(2.9333\) | |
| 176400.lf4 | 176400kd4 | \([0, 0, 0, 12932325, -42200061750]\) | \(1613964717/6125000\) | \(-907748624664000000000000\) | \([2]\) | \(15925248\) | \(3.2799\) |
Rank
sage: E.rank()
The elliptic curves in class 176400kd have rank \(1\).
Complex multiplication
The elliptic curves in class 176400kd do not have complex multiplication.Modular form 176400.2.a.kd
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.
