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SageMath
E = EllipticCurve("fk1")
E.isogeny_class()
Elliptic curves in class 424830fk
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
424830.fk6 | 424830fk1 | \([1, 1, 1, -1133175, -570931635]\) | \(-56667352321/16711680\) | \(-47457174689982382080\) | \([2]\) | \(14155776\) | \(2.4900\) | \(\Gamma_0(N)\)-optimal* |
424830.fk5 | 424830fk2 | \([1, 1, 1, -19259255, -32538086323]\) | \(278202094583041/16646400\) | \(47271795101349638400\) | \([2, 2]\) | \(28311552\) | \(2.8366\) | \(\Gamma_0(N)\)-optimal* |
424830.fk4 | 424830fk3 | \([1, 1, 1, -20392135, -28496423635]\) | \(330240275458561/67652010000\) | \(192115529779078764810000\) | \([2, 2]\) | \(56623104\) | \(3.1832\) | \(\Gamma_0(N)\)-optimal* |
424830.fk2 | 424830fk4 | \([1, 1, 1, -308143655, -2082115127443]\) | \(1139466686381936641/4080\) | \(11586224289546480\) | \([2]\) | \(56623104\) | \(3.1832\) | |
424830.fk3 | 424830fk5 | \([1, 1, 1, -102242715, 372604158597]\) | \(41623544884956481/2962701562500\) | \(8413363923067355076562500\) | \([2, 2]\) | \(113246208\) | \(3.5297\) | \(\Gamma_0(N)\)-optimal* |
424830.fk7 | 424830fk6 | \([1, 1, 1, 43332365, -170907936235]\) | \(3168685387909439/6278181696900\) | \(-17828534625198263906328900\) | \([2]\) | \(113246208\) | \(3.5297\) | |
424830.fk1 | 424830fk7 | \([1, 1, 1, -1606848965, 24791159911097]\) | \(161572377633716256481/914742821250\) | \(2597650856435054901521250\) | \([2]\) | \(226492416\) | \(3.8763\) | \(\Gamma_0(N)\)-optimal* |
424830.fk8 | 424830fk8 | \([1, 1, 1, 92754255, 1626122680545]\) | \(31077313442863199/420227050781250\) | \(-1193344329138774719238281250\) | \([2]\) | \(226492416\) | \(3.8763\) |
Rank
sage: E.rank()
The elliptic curves in class 424830fk have rank \(0\).
Complex multiplication
The elliptic curves in class 424830fk do not have complex multiplication.Modular form 424830.2.a.fk
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 4 & 8 & 8 & 16 & 16 \\ 2 & 1 & 2 & 2 & 4 & 4 & 8 & 8 \\ 4 & 2 & 1 & 4 & 2 & 2 & 4 & 4 \\ 4 & 2 & 4 & 1 & 8 & 8 & 16 & 16 \\ 8 & 4 & 2 & 8 & 1 & 4 & 2 & 2 \\ 8 & 4 & 2 & 8 & 4 & 1 & 8 & 8 \\ 16 & 8 & 4 & 16 & 2 & 8 & 1 & 4 \\ 16 & 8 & 4 & 16 & 2 & 8 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.