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SageMath
E = EllipticCurve("hh1")
E.isogeny_class()
Elliptic curves in class 193200.hh
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 193200.hh1 | 193200br4 | \([0, 1, 0, -1717408, -866852812]\) | \(8753151307882969/65205\) | \(4173120000000\) | \([2]\) | \(2162688\) | \(2.0162\) | |
| 193200.hh2 | 193200br2 | \([0, 1, 0, -107408, -13552812]\) | \(2141202151369/5832225\) | \(373262400000000\) | \([2, 2]\) | \(1081344\) | \(1.6696\) | |
| 193200.hh3 | 193200br3 | \([0, 1, 0, -65408, -24220812]\) | \(-483551781049/3672913125\) | \(-235066440000000000\) | \([2]\) | \(2162688\) | \(2.0162\) | |
| 193200.hh4 | 193200br1 | \([0, 1, 0, -9408, -28812]\) | \(1439069689/828345\) | \(53014080000000\) | \([2]\) | \(540672\) | \(1.3231\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 193200.hh have rank \(1\).
Complex multiplication
The elliptic curves in class 193200.hh do not have complex multiplication.Modular form 193200.2.a.hh
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
