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SageMath
E = EllipticCurve("n1")
E.isogeny_class()
Elliptic curves in class 274890.n
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
274890.n1 | 274890n7 | \([1, 1, 0, -9860587643, 376875213421533]\) | \(901247067798311192691198986281/552431869440\) | \(64993057007746560\) | \([2]\) | \(191102976\) | \(3.9337\) | |
274890.n2 | 274890n8 | \([1, 1, 0, -620426363, 5805359562717]\) | \(224494757451893010998773801/6152490825146276160000\) | \(723834393087634243947840000\) | \([2]\) | \(191102976\) | \(3.9337\) | |
274890.n3 | 274890n6 | \([1, 1, 0, -616286843, 5888480296413]\) | \(220031146443748723000125481/172266701724057600\) | \(20267005191133652582400\) | \([2, 2]\) | \(95551488\) | \(3.5872\) | |
274890.n4 | 274890n4 | \([1, 1, 0, -121760468, 516709162488]\) | \(1696892787277117093383481/1440538624914939000\) | \(169477928682617658411000\) | \([2]\) | \(63700992\) | \(3.3844\) | |
274890.n5 | 274890n5 | \([1, 1, 0, -79741988, -271185960408]\) | \(476646772170172569823801/5862293314453125000\) | \(689692946152095703125000\) | \([2]\) | \(63700992\) | \(3.3844\) | |
274890.n6 | 274890n3 | \([1, 1, 0, -38259323, 93291986397]\) | \(-52643812360427830814761/1504091705903677440\) | \(-176954885107861747138560\) | \([2]\) | \(47775744\) | \(3.2406\) | |
274890.n7 | 274890n2 | \([1, 1, 0, -9305468, 4206745488]\) | \(757443433548897303481/373234243041000000\) | \(43910635459530609000000\) | \([2, 2]\) | \(31850496\) | \(3.0379\) | |
274890.n8 | 274890n1 | \([1, 1, 0, 2125252, 505478352]\) | \(9023321954633914439/6156756739584000\) | \(-724336273655318016000\) | \([2]\) | \(15925248\) | \(2.6913\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 274890.n have rank \(2\).
Complex multiplication
The elliptic curves in class 274890.n do not have complex multiplication.Modular form 274890.2.a.n
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 4 & 2 & 3 & 12 & 4 & 6 & 12 \\ 4 & 1 & 2 & 12 & 3 & 4 & 6 & 12 \\ 2 & 2 & 1 & 6 & 6 & 2 & 3 & 6 \\ 3 & 12 & 6 & 1 & 4 & 12 & 2 & 4 \\ 12 & 3 & 6 & 4 & 1 & 12 & 2 & 4 \\ 4 & 4 & 2 & 12 & 12 & 1 & 6 & 3 \\ 6 & 6 & 3 & 2 & 2 & 6 & 1 & 2 \\ 12 & 12 & 6 & 4 & 4 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.