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SageMath
E = EllipticCurve("er1")
E.isogeny_class()
Elliptic curves in class 22050er
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 22050.df6 | 22050er1 | \([1, -1, 1, 110020, -19684353]\) | \(109902239/188160\) | \(-252152395740000000\) | \([2]\) | \(294912\) | \(2.0240\) | \(\Gamma_0(N)\)-optimal |
| 22050.df5 | 22050er2 | \([1, -1, 1, -771980, -203140353]\) | \(37966934881/8643600\) | \(11583250679306250000\) | \([2, 2]\) | \(589824\) | \(2.3706\) | |
| 22050.df4 | 22050er3 | \([1, -1, 1, -4079480, 2998519647]\) | \(5602762882081/345888060\) | \(463523081350238437500\) | \([2]\) | \(1179648\) | \(2.7172\) | |
| 22050.df2 | 22050er4 | \([1, -1, 1, -11576480, -15156568353]\) | \(128031684631201/9922500\) | \(13297098994101562500\) | \([2, 2]\) | \(1179648\) | \(2.7172\) | |
| 22050.df3 | 22050er5 | \([1, -1, 1, -10804730, -17264989353]\) | \(-104094944089921/35880468750\) | \(-48083259755456542968750\) | \([2]\) | \(2359296\) | \(3.0637\) | |
| 22050.df1 | 22050er6 | \([1, -1, 1, -185220230, -970197193353]\) | \(524388516989299201/3150\) | \(4221301267968750\) | \([2]\) | \(2359296\) | \(3.0637\) |
Rank
sage: E.rank()
The elliptic curves in class 22050er have rank \(0\).
Complex multiplication
The elliptic curves in class 22050er do not have complex multiplication.Modular form 22050.2.a.er
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 Cremona 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 Cremona labels.
