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SageMath
E = EllipticCurve("id1")
E.isogeny_class()
Elliptic curves in class 369600id
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
369600.id7 | 369600id1 | \([0, -1, 0, -1876033, -449036063]\) | \(178272935636041/81841914000\) | \(335224479744000000000\) | \([2]\) | \(10616832\) | \(2.6333\) | \(\Gamma_0(N)\)-optimal |
369600.id5 | 369600id2 | \([0, -1, 0, -25204033, -48668012063]\) | \(432288716775559561/270140062500\) | \(1106493696000000000000\) | \([2, 2]\) | \(21233664\) | \(2.9798\) | |
369600.id4 | 369600id3 | \([0, -1, 0, -76396033, 257024043937]\) | \(12038605770121350841/757333463040\) | \(3102037864611840000000\) | \([2]\) | \(31850496\) | \(3.1826\) | |
369600.id6 | 369600id4 | \([0, -1, 0, -20452033, -67585724063]\) | \(-230979395175477481/348191894531250\) | \(-1426194000000000000000000\) | \([2]\) | \(42467328\) | \(3.3264\) | |
369600.id2 | 369600id5 | \([0, -1, 0, -403204033, -3116138012063]\) | \(1769857772964702379561/691787250\) | \(2833560576000000000\) | \([2]\) | \(42467328\) | \(3.3264\) | |
369600.id3 | 369600id6 | \([0, -1, 0, -81004033, 224274987937]\) | \(14351050585434661561/3001282273281600\) | \(12293252191361433600000000\) | \([2, 2]\) | \(63700992\) | \(3.5291\) | |
369600.id8 | 369600id7 | \([0, -1, 0, 174547967, 1353559275937]\) | \(143584693754978072519/276341298967965000\) | \(-1131893960572784640000000000\) | \([2]\) | \(127401984\) | \(3.8757\) | |
369600.id1 | 369600id8 | \([0, -1, 0, -410284033, -3001022612063]\) | \(1864737106103260904761/129177711985836360\) | \(529111908293985730560000000\) | \([2]\) | \(127401984\) | \(3.8757\) |
Rank
sage: E.rank()
The elliptic curves in class 369600id have rank \(1\).
Complex multiplication
The elliptic curves in class 369600id do not have complex multiplication.Modular form 369600.2.a.id
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.