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SageMath
sage: E = EllipticCurve("r1")
sage: E.isogeny_class()
Elliptic curves in class 72450.r
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 72450.r1 | 72450be4 | \([1, -1, 0, -678298167, -18483604259]\) | \(3029968325354577848895529/1753440696000000000000\) | \(19972785427875000000000000000\) | \([2]\) | \(53084160\) | \(4.1198\) | |
| 72450.r2 | 72450be2 | \([1, -1, 0, -466614792, -3879452132384]\) | \(986396822567235411402169/6336721794060000\) | \(72179221685464687500000\) | \([2]\) | \(17694720\) | \(3.5705\) | |
| 72450.r3 | 72450be1 | \([1, -1, 0, -28602792, -63053576384]\) | \(-227196402372228188089/19338934824115200\) | \(-220282554480937200000000\) | \([2]\) | \(8847360\) | \(3.2239\) | \(\Gamma_0(N)\)-optimal |
| 72450.r4 | 72450be3 | \([1, -1, 0, 169573833, -2374036259]\) | \(47342661265381757089751/27397579603968000000\) | \(-312075555176448000000000000\) | \([2]\) | \(26542080\) | \(3.7732\) |
Rank
sage: E.rank()
The elliptic curves in class 72450.r have rank \(1\).
Complex multiplication
The elliptic curves in class 72450.r do not have complex multiplication.Modular form 72450.2.a.r
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.
\(\left(\begin{array}{rrrr} 1 & 3 & 6 & 2 \\ 3 & 1 & 2 & 6 \\ 6 & 2 & 1 & 3 \\ 2 & 6 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
