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SageMath
E = EllipticCurve("gg1")
E.isogeny_class()
Elliptic curves in class 73920gg
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 73920.eo7 | 73920gg1 | \([0, 1, 0, -75041, -3622305]\) | \(178272935636041/81841914000\) | \(21454366703616000\) | \([2]\) | \(442368\) | \(1.8285\) | \(\Gamma_0(N)\)-optimal |
| 73920.eo5 | 73920gg2 | \([0, 1, 0, -1008161, -389747361]\) | \(432288716775559561/270140062500\) | \(70815596544000000\) | \([2, 2]\) | \(884736\) | \(2.1751\) | |
| 73920.eo4 | 73920gg3 | \([0, 1, 0, -3055841, 2054970015]\) | \(12038605770121350841/757333463040\) | \(198530423335157760\) | \([2]\) | \(1327104\) | \(2.3779\) | |
| 73920.eo6 | 73920gg4 | \([0, 1, 0, -818081, -541013025]\) | \(-230979395175477481/348191894531250\) | \(-91276416000000000000\) | \([2]\) | \(1769472\) | \(2.5217\) | |
| 73920.eo2 | 73920gg5 | \([0, 1, 0, -16128161, -24935555361]\) | \(1769857772964702379561/691787250\) | \(181347876864000\) | \([2]\) | \(1769472\) | \(2.5217\) | |
| 73920.eo3 | 73920gg6 | \([0, 1, 0, -3240161, 1792903839]\) | \(14351050585434661561/3001282273281600\) | \(786768140247131750400\) | \([2, 2]\) | \(2654208\) | \(2.7244\) | |
| 73920.eo8 | 73920gg7 | \([0, 1, 0, 6981919, 10831266975]\) | \(143584693754978072519/276341298967965000\) | \(-72441213476658216960000\) | \([2]\) | \(5308416\) | \(3.0710\) | |
| 73920.eo1 | 73920gg8 | \([0, 1, 0, -16411361, -24014745441]\) | \(1864737106103260904761/129177711985836360\) | \(33863162130815086755840\) | \([2]\) | \(5308416\) | \(3.0710\) |
Rank
sage: E.rank()
The elliptic curves in class 73920gg have rank \(1\).
Complex multiplication
The elliptic curves in class 73920gg do not have complex multiplication.Modular form 73920.2.a.gg
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 3 & 4 & 4 & 6 & 12 & 12 \\ 2 & 1 & 6 & 2 & 2 & 3 & 6 & 6 \\ 3 & 6 & 1 & 12 & 12 & 2 & 4 & 4 \\ 4 & 2 & 12 & 1 & 4 & 6 & 3 & 12 \\ 4 & 2 & 12 & 4 & 1 & 6 & 12 & 3 \\ 6 & 3 & 2 & 6 & 6 & 1 & 2 & 2 \\ 12 & 6 & 4 & 3 & 12 & 2 & 1 & 4 \\ 12 & 6 & 4 & 12 & 3 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
