Show commands:
SageMath
E = EllipticCurve("cm1")
E.isogeny_class()
Elliptic curves in class 13950.cm
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 13950.cm1 | 13950cf5 | \([1, -1, 1, -69192005, 221547010497]\) | \(3216206300355197383681/57660\) | \(656783437500\) | \([4]\) | \(786432\) | \(2.7359\) | |
| 13950.cm2 | 13950cf4 | \([1, -1, 1, -4324505, 3462475497]\) | \(785209010066844481/3324675600\) | \(37870133006250000\) | \([2, 2]\) | \(393216\) | \(2.3893\) | |
| 13950.cm3 | 13950cf6 | \([1, -1, 1, -4257005, 3575740497]\) | \(-749011598724977281/51173462246460\) | \(-582897718401083437500\) | \([2]\) | \(786432\) | \(2.7359\) | |
| 13950.cm4 | 13950cf3 | \([1, -1, 1, -832505, -227740503]\) | \(5601911201812801/1271193750000\) | \(14479691308593750000\) | \([2]\) | \(393216\) | \(2.3893\) | |
| 13950.cm5 | 13950cf2 | \([1, -1, 1, -274505, 52375497]\) | \(200828550012481/12454560000\) | \(141865222500000000\) | \([2, 2]\) | \(196608\) | \(2.0427\) | |
| 13950.cm6 | 13950cf1 | \([1, -1, 1, 13495, 3415497]\) | \(23862997439/457113600\) | \(-5206809600000000\) | \([2]\) | \(98304\) | \(1.6961\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 13950.cm have rank \(1\).
Complex multiplication
The elliptic curves in class 13950.cm do not have complex multiplication.Modular form 13950.2.a.cm
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 8 & 4 & 8 \\ 2 & 1 & 2 & 4 & 2 & 4 \\ 4 & 2 & 1 & 8 & 4 & 8 \\ 8 & 4 & 8 & 1 & 2 & 4 \\ 4 & 2 & 4 & 2 & 1 & 2 \\ 8 & 4 & 8 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
