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SageMath
E = EllipticCurve("p1")
E.isogeny_class()
Elliptic curves in class 2070.p
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 2070.p1 | 2070m4 | \([1, -1, 1, -35372, -2273129]\) | \(248656466619387/29607177800\) | \(582758080637400\) | \([2]\) | \(10368\) | \(1.5643\) | |
| 2070.p2 | 2070m3 | \([1, -1, 1, -34292, -2435561]\) | \(226568219476347/3893440\) | \(76634579520\) | \([2]\) | \(5184\) | \(1.2177\) | |
| 2070.p3 | 2070m2 | \([1, -1, 1, -8372, 296471]\) | \(2403250125069123/4232000000\) | \(114264000000\) | \([6]\) | \(3456\) | \(1.0150\) | |
| 2070.p4 | 2070m1 | \([1, -1, 1, -692, 1559]\) | \(1355469437763/753664000\) | \(20348928000\) | \([6]\) | \(1728\) | \(0.66840\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 2070.p have rank \(1\).
Complex multiplication
The elliptic curves in class 2070.p do not have complex multiplication.Modular form 2070.2.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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
