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SageMath
E = EllipticCurve("c1")
E.isogeny_class()
Elliptic curves in class 210.c
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 210.c1 | 210c5 | \([1, 1, 1, -16800, -845133]\) | \(524388516989299201/3150\) | \(3150\) | \([2]\) | \(256\) | \(0.73676\) | |
| 210.c2 | 210c3 | \([1, 1, 1, -1050, -13533]\) | \(128031684631201/9922500\) | \(9922500\) | \([2, 2]\) | \(128\) | \(0.39018\) | |
| 210.c3 | 210c6 | \([1, 1, 1, -980, -15325]\) | \(-104094944089921/35880468750\) | \(-35880468750\) | \([2]\) | \(256\) | \(0.73676\) | |
| 210.c4 | 210c4 | \([1, 1, 1, -370, 2435]\) | \(5602762882081/345888060\) | \(345888060\) | \([4]\) | \(128\) | \(0.39018\) | |
| 210.c5 | 210c2 | \([1, 1, 1, -70, -205]\) | \(37966934881/8643600\) | \(8643600\) | \([2, 4]\) | \(64\) | \(0.043610\) | |
| 210.c6 | 210c1 | \([1, 1, 1, 10, -13]\) | \(109902239/188160\) | \(-188160\) | \([4]\) | \(32\) | \(-0.30296\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 210.c have rank \(0\).
Complex multiplication
The elliptic curves in class 210.c do not have complex multiplication.Modular form 210.2.a.c
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 8 & 4 & 8 \\ 2 & 1 & 2 & 4 & 2 & 4 \\ 4 & 2 & 1 & 8 & 4 & 8 \\ 8 & 4 & 8 & 1 & 2 & 4 \\ 4 & 2 & 4 & 2 & 1 & 2 \\ 8 & 4 & 8 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
