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SageMath
E = EllipticCurve("dz1")
E.isogeny_class()
Elliptic curves in class 20160.dz
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 20160.dz1 | 20160ca4 | \([0, 0, 0, -3024012, 2024059984]\) | \(128025588102048008/7875\) | \(188116992000\) | \([2]\) | \(196608\) | \(2.0718\) | |
| 20160.dz2 | 20160ca3 | \([0, 0, 0, -211692, 23556976]\) | \(43919722445768/15380859375\) | \(367416000000000000\) | \([2]\) | \(196608\) | \(2.0718\) | |
| 20160.dz3 | 20160ca2 | \([0, 0, 0, -189012, 31621984]\) | \(250094631024064/62015625\) | \(185177664000000\) | \([2, 2]\) | \(98304\) | \(1.7253\) | |
| 20160.dz4 | 20160ca1 | \([0, 0, 0, -10407, 616156]\) | \(-2671731885376/1969120125\) | \(-91871268552000\) | \([2]\) | \(49152\) | \(1.3787\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 20160.dz have rank \(1\).
Complex multiplication
The elliptic curves in class 20160.dz do not have complex multiplication.Modular form 20160.2.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.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
