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SageMath
E = EllipticCurve("hk1")
E.isogeny_class()
Elliptic curves in class 323400.hk
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 323400.hk1 | 323400hk4 | \([0, 1, 0, -862808, -308762112]\) | \(37736227588/33\) | \(62118672000000\) | \([2]\) | \(3538944\) | \(1.9470\) | |
| 323400.hk2 | 323400hk3 | \([0, 1, 0, -127808, 10815888]\) | \(122657188/43923\) | \(82679952432000000\) | \([2]\) | \(3538944\) | \(1.9470\) | |
| 323400.hk3 | 323400hk2 | \([0, 1, 0, -54308, -4766112]\) | \(37642192/1089\) | \(512479044000000\) | \([2, 2]\) | \(1769472\) | \(1.6004\) | |
| 323400.hk4 | 323400hk1 | \([0, 1, 0, 817, -245862]\) | \(2048/891\) | \(-26206314750000\) | \([2]\) | \(884736\) | \(1.2538\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 323400.hk have rank \(0\).
Complex multiplication
The elliptic curves in class 323400.hk do not have complex multiplication.Modular form 323400.2.a.hk
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.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
