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SageMath
E = EllipticCurve("r1")
E.isogeny_class()
Elliptic curves in class 111090r
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 111090.u7 | 111090r1 | \([1, 0, 1, -263189, -51971488]\) | \(13619385906841/6048000\) | \(895321056672000\) | \([2]\) | \(1216512\) | \(1.8276\) | \(\Gamma_0(N)\)-optimal |
| 111090.u6 | 111090r2 | \([1, 0, 1, -305509, -34146304]\) | \(21302308926361/8930250000\) | \(1321997497742250000\) | \([2, 2]\) | \(2433024\) | \(2.1741\) | |
| 111090.u5 | 111090r3 | \([1, 0, 1, -778964, 201043922]\) | \(353108405631241/86318776320\) | \(12778276789923348480\) | \([2]\) | \(3649536\) | \(2.3769\) | |
| 111090.u8 | 111090r4 | \([1, 0, 1, 1016991, -250507304]\) | \(785793873833639/637994920500\) | \(-94446145233701824500\) | \([2]\) | \(4866048\) | \(2.5207\) | |
| 111090.u4 | 111090r5 | \([1, 0, 1, -2305129, 1323195752]\) | \(9150443179640281/184570312500\) | \(27323030293945312500\) | \([2]\) | \(4866048\) | \(2.5207\) | |
| 111090.u2 | 111090r6 | \([1, 0, 1, -11612884, 15229857746]\) | \(1169975873419524361/108425318400\) | \(16050838399452057600\) | \([2, 2]\) | \(7299072\) | \(2.7234\) | |
| 111090.u3 | 111090r7 | \([1, 0, 1, -10766484, 17544253906]\) | \(-932348627918877961/358766164249920\) | \(-53110268067856925378880\) | \([2]\) | \(14598144\) | \(3.0700\) | |
| 111090.u1 | 111090r8 | \([1, 0, 1, -185802004, 974802882002]\) | \(4791901410190533590281/41160000\) | \(6093157191240000\) | \([2]\) | \(14598144\) | \(3.0700\) |
Rank
sage: E.rank()
The elliptic curves in class 111090r have rank \(1\).
Complex multiplication
The elliptic curves in class 111090r do not have complex multiplication.Modular form 111090.2.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 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.
