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SageMath
E = EllipticCurve("dg1")
E.isogeny_class()
Elliptic curves in class 123840.dg
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 123840.dg1 | 123840cf4 | \([0, 0, 0, -9353388, -11010151888]\) | \(947094050118111698/20769216075\) | \(1984529500559769600\) | \([2]\) | \(5242880\) | \(2.6267\) | |
| 123840.dg2 | 123840cf2 | \([0, 0, 0, -605388, -159132688]\) | \(513591322675396/68238500625\) | \(3260145136803840000\) | \([2, 2]\) | \(2621440\) | \(2.2801\) | |
| 123840.dg3 | 123840cf1 | \([0, 0, 0, -155388, 21047312]\) | \(34739908901584/4081640625\) | \(48750854400000000\) | \([2]\) | \(1310720\) | \(1.9336\) | \(\Gamma_0(N)\)-optimal |
| 123840.dg4 | 123840cf3 | \([0, 0, 0, 942612, -839633488]\) | \(969360123836302/3748293231075\) | \(-358154995689544089600\) | \([2]\) | \(5242880\) | \(2.6267\) |
Rank
sage: E.rank()
The elliptic curves in class 123840.dg have rank \(1\).
Complex multiplication
The elliptic curves in class 123840.dg do not have complex multiplication.Modular form 123840.2.a.dg
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.
