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SageMath
E = EllipticCurve("p1")
E.isogeny_class()
Elliptic curves in class 84270.p
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 84270.p1 | 84270k4 | \([1, 0, 1, -63525594, -194887057988]\) | \(1279130011356875761/27818640\) | \(616582383077644560\) | \([2]\) | \(11501568\) | \(2.9402\) | |
| 84270.p2 | 84270k2 | \([1, 0, 1, -3974794, -3038200708]\) | \(313337384670961/1456185600\) | \(32275423509249542400\) | \([2, 2]\) | \(5750784\) | \(2.5936\) | |
| 84270.p3 | 84270k3 | \([1, 0, 1, -1952314, -6126932164]\) | \(-37129335824881/710143290000\) | \(-15739872332896174410000\) | \([2]\) | \(11501568\) | \(2.9402\) | |
| 84270.p4 | 84270k1 | \([1, 0, 1, -379274, 7923836]\) | \(272223782641/156303360\) | \(3464364116716093440\) | \([2]\) | \(2875392\) | \(2.2471\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 84270.p have rank \(0\).
Complex multiplication
The elliptic curves in class 84270.p do not have complex multiplication.Modular form 84270.2.a.p
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.
