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SageMath
E = EllipticCurve("dv1")
E.isogeny_class()
Elliptic curves in class 489762dv
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
489762.dv4 | 489762dv1 | \([1, -1, 1, -911111, 546264047]\) | \(-23771111713777/22848457968\) | \(-80397868923370737648\) | \([2]\) | \(17694720\) | \(2.5156\) | \(\Gamma_0(N)\)-optimal* |
489762.dv3 | 489762dv2 | \([1, -1, 1, -17003291, 26982497351]\) | \(154502321244119857/55101928644\) | \(193889567635142190084\) | \([2, 2]\) | \(35389440\) | \(2.8622\) | \(\Gamma_0(N)\)-optimal* |
489762.dv1 | 489762dv3 | \([1, -1, 1, -272029361, 1726986279971]\) | \(632678989847546725777/80515134\) | \(283312125428578974\) | \([2]\) | \(70778880\) | \(3.2088\) | \(\Gamma_0(N)\)-optimal* |
489762.dv2 | 489762dv4 | \([1, -1, 1, -19452101, 18704540027]\) | \(231331938231569617/90942310746882\) | \(320002688551514287703202\) | \([2]\) | \(70778880\) | \(3.2088\) |
Rank
sage: E.rank()
The elliptic curves in class 489762dv have rank \(0\).
Complex multiplication
The elliptic curves in class 489762dv do not have complex multiplication.Modular form 489762.2.a.dv
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.