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SageMath
E = EllipticCurve("b1")
E.isogeny_class()
Elliptic curves in class 51870b
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
51870.e4 | 51870b1 | \([1, 1, 0, -14213, -690483]\) | \(-317562142497484249/18670942617600\) | \(-18670942617600\) | \([2]\) | \(163840\) | \(1.3024\) | \(\Gamma_0(N)\)-optimal |
51870.e3 | 51870b2 | \([1, 1, 0, -230533, -42699827]\) | \(1354958399265695661529/4304795040000\) | \(4304795040000\) | \([2, 2]\) | \(327680\) | \(1.6490\) | |
51870.e2 | 51870b3 | \([1, 1, 0, -233653, -41488643]\) | \(1410719602237262088409/76269550743750000\) | \(76269550743750000\) | \([2]\) | \(655360\) | \(1.9955\) | |
51870.e1 | 51870b4 | \([1, 1, 0, -3688533, -2728182627]\) | \(5549896908024170183373529/56019600\) | \(56019600\) | \([2]\) | \(655360\) | \(1.9955\) |
Rank
sage: E.rank()
The elliptic curves in class 51870b have rank \(0\).
Complex multiplication
The elliptic curves in class 51870b do not have complex multiplication.Modular form 51870.2.a.b
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.