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SageMath
E = EllipticCurve("i1")
E.isogeny_class()
Elliptic curves in class 84474.i
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
84474.i1 | 84474p4 | \([1, -1, 0, -67382703, 212914421181]\) | \(986551739719628473/111045168\) | \(3808454746568342832\) | \([2]\) | \(7741440\) | \(2.9896\) | |
84474.i2 | 84474p3 | \([1, -1, 0, -7601103, -2733821091]\) | \(1416134368422073/725251155408\) | \(24873537993726773072592\) | \([2]\) | \(7741440\) | \(2.9896\) | |
84474.i3 | 84474p2 | \([1, -1, 0, -4222143, 3309786765]\) | \(242702053576633/2554695936\) | \(87616994406258680064\) | \([2, 2]\) | \(3870720\) | \(2.6430\) | |
84474.i4 | 84474p1 | \([1, -1, 0, -63423, 128365965]\) | \(-822656953/207028224\) | \(-7100332563470155776\) | \([2]\) | \(1935360\) | \(2.2965\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 84474.i have rank \(1\).
Complex multiplication
The elliptic curves in class 84474.i do not have complex multiplication.Modular form 84474.2.a.i
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.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.