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SageMath
E = EllipticCurve("dz1")
E.isogeny_class()
Elliptic curves in class 72450.dz
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 72450.dz1 | 72450ej6 | \([1, -1, 1, -50390424005, 4353829215043497]\) | \(1242282009445982549834550082561/41992020\) | \(478315352812500\) | \([2]\) | \(75497472\) | \(4.2365\) | |
| 72450.dz2 | 72450ej4 | \([1, -1, 1, -3149401505, 68029171798497]\) | \(303291507481995500913332161/1763329743680400\) | \(20085427861609556250000\) | \([2, 2]\) | \(37748736\) | \(3.8899\) | |
| 72450.dz3 | 72450ej5 | \([1, -1, 1, -3147579005, 68111836753497]\) | \(-302765284673144739899429761/731344538939408411220\) | \(-8330471388856698934052812500\) | \([2]\) | \(75497472\) | \(4.2365\) | |
| 72450.dz4 | 72450ej2 | \([1, -1, 1, -196951505, 1061700898497]\) | \(74174404299602673044161/178530248806560000\) | \(2033571115312222500000000\) | \([2, 2]\) | \(18874368\) | \(3.5434\) | |
| 72450.dz5 | 72450ej3 | \([1, -1, 1, -124501505, 1852709998497]\) | \(-18736995756767139956161/119334500162058560400\) | \(-1359294540908448289556250000\) | \([2]\) | \(37748736\) | \(3.8899\) | |
| 72450.dz6 | 72450ej1 | \([1, -1, 1, -16951505, 2940898497]\) | \(47293441677949844161/27041817600000000\) | \(308023203600000000000000\) | \([2]\) | \(9437184\) | \(3.1968\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 72450.dz have rank \(0\).
Complex multiplication
The elliptic curves in class 72450.dz do not have complex multiplication.Modular form 72450.2.a.dz
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 8 & 8 \\ 4 & 2 & 4 & 1 & 2 & 2 \\ 8 & 4 & 8 & 2 & 1 & 4 \\ 8 & 4 & 8 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
