Show commands:
SageMath
E = EllipticCurve("dg1")
E.isogeny_class()
Elliptic curves in class 127050.dg
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 127050.dg1 | 127050cr4 | \([1, 0, 1, -278313676, -1787127468502]\) | \(86129359107301290313/9166294368\) | \(253728900263569500000\) | \([2]\) | \(29491200\) | \(3.3430\) | |
| 127050.dg2 | 127050cr2 | \([1, 0, 1, -17437676, -27779724502]\) | \(21184262604460873/216872764416\) | \(6003177053149584000000\) | \([2, 2]\) | \(14745600\) | \(2.9964\) | |
| 127050.dg3 | 127050cr3 | \([1, 0, 1, -4369676, -68447340502]\) | \(-333345918055753/72923718045024\) | \(-2018575232243136913500000\) | \([2]\) | \(29491200\) | \(3.3430\) | |
| 127050.dg4 | 127050cr1 | \([1, 0, 1, -1949676, 346483498]\) | \(29609739866953/15259926528\) | \(422404542185472000000\) | \([2]\) | \(7372800\) | \(2.6498\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 127050.dg have rank \(1\).
Complex multiplication
The elliptic curves in class 127050.dg do not have complex multiplication.Modular form 127050.2.a.dg
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.
