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SageMath
E = EllipticCurve("p1")
E.isogeny_class()
Elliptic curves in class 37030p
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 37030.r3 | 37030p1 | \([1, -1, 1, -157477, 12123101]\) | \(2917464019569/1262240000\) | \(186856820531360000\) | \([4]\) | \(405504\) | \(2.0103\) | \(\Gamma_0(N)\)-optimal |
| 37030.r2 | 37030p2 | \([1, -1, 1, -1215477, -507143299]\) | \(1341518286067569/24894528400\) | \(3685283642929747600\) | \([2, 2]\) | \(811008\) | \(2.3569\) | |
| 37030.r4 | 37030p3 | \([1, -1, 1, 1223, -1477096539]\) | \(1367631/6366992112460\) | \(-942543337624004076940\) | \([2]\) | \(1622016\) | \(2.7035\) | |
| 37030.r1 | 37030p4 | \([1, -1, 1, -19360177, -32782935659]\) | \(5421065386069310769/1919709260\) | \(284185866925632140\) | \([2]\) | \(1622016\) | \(2.7035\) |
Rank
sage: E.rank()
The elliptic curves in class 37030p have rank \(0\).
Complex multiplication
The elliptic curves in class 37030p do not have complex multiplication.Modular form 37030.2.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 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.
