Show commands:
SageMath
E = EllipticCurve("a1")
E.isogeny_class()
Elliptic curves in class 2070a
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 2070.a4 | 2070a1 | \([1, -1, 0, -3810, 91476]\) | \(226568219476347/3893440\) | \(105122880\) | \([6]\) | \(1728\) | \(0.66840\) | \(\Gamma_0(N)\)-optimal |
| 2070.a3 | 2070a2 | \([1, -1, 0, -3930, 85500]\) | \(248656466619387/29607177800\) | \(799393800600\) | \([6]\) | \(3456\) | \(1.0150\) | |
| 2070.a2 | 2070a3 | \([1, -1, 0, -6225, -35875]\) | \(1355469437763/753664000\) | \(14834368512000\) | \([2]\) | \(5184\) | \(1.2177\) | |
| 2070.a1 | 2070a4 | \([1, -1, 0, -75345, -7929379]\) | \(2403250125069123/4232000000\) | \(83298456000000\) | \([2]\) | \(10368\) | \(1.5643\) |
Rank
sage: E.rank()
The elliptic curves in class 2070a have rank \(0\).
Complex multiplication
The elliptic curves in class 2070a do not have complex multiplication.Modular form 2070.2.a.a
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
