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SageMath
E = EllipticCurve("gi1")
E.isogeny_class()
Elliptic curves in class 80850.gi
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 80850.gi1 | 80850fw8 | \([1, 0, 0, -314123713, -2010568491583]\) | \(1864737106103260904761/129177711985836360\) | \(237462947459713467463125000\) | \([2]\) | \(31850496\) | \(3.8090\) | |
| 80850.gi2 | 80850fw5 | \([1, 0, 0, -308703088, -2087684783458]\) | \(1769857772964702379561/691787250\) | \(1271688721488281250\) | \([2]\) | \(10616832\) | \(3.2596\) | |
| 80850.gi3 | 80850fw6 | \([1, 0, 0, -62018713, 150223463417]\) | \(14351050585434661561/3001282273281600\) | \(5517154033895421225000000\) | \([2, 2]\) | \(15925248\) | \(3.4624\) | |
| 80850.gi4 | 80850fw3 | \([1, 0, 0, -58490713, 172164095417]\) | \(12038605770121350841/757333463040\) | \(1392180071768640000000\) | \([4]\) | \(7962624\) | \(3.1158\) | |
| 80850.gi5 | 80850fw2 | \([1, 0, 0, -19296838, -32611002208]\) | \(432288716775559561/270140062500\) | \(496589190829101562500\) | \([2, 2]\) | \(5308416\) | \(2.9131\) | |
| 80850.gi6 | 80850fw4 | \([1, 0, 0, -15658588, -45283026958]\) | \(-230979395175477481/348191894531250\) | \(-640069190620422363281250\) | \([2]\) | \(10616832\) | \(3.2596\) | |
| 80850.gi7 | 80850fw1 | \([1, 0, 0, -1436338, -301357708]\) | \(178272935636041/81841914000\) | \(150447177190406250000\) | \([4]\) | \(2654208\) | \(2.5665\) | \(\Gamma_0(N)\)-optimal |
| 80850.gi8 | 80850fw7 | \([1, 0, 0, 133638287, 906829082417]\) | \(143584693754978072519/276341298967965000\) | \(-507988710660658035703125000\) | \([2]\) | \(31850496\) | \(3.8090\) |
Rank
sage: E.rank()
The elliptic curves in class 80850.gi have rank \(0\).
Complex multiplication
The elliptic curves in class 80850.gi do not have complex multiplication.Modular form 80850.2.a.gi
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 3 & 2 & 4 & 6 & 12 & 12 & 4 \\ 3 & 1 & 6 & 12 & 2 & 4 & 4 & 12 \\ 2 & 6 & 1 & 2 & 3 & 6 & 6 & 2 \\ 4 & 12 & 2 & 1 & 6 & 12 & 3 & 4 \\ 6 & 2 & 3 & 6 & 1 & 2 & 2 & 6 \\ 12 & 4 & 6 & 12 & 2 & 1 & 4 & 3 \\ 12 & 4 & 6 & 3 & 2 & 4 & 1 & 12 \\ 4 & 12 & 2 & 4 & 6 & 3 & 12 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
