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SageMath
E = EllipticCurve("i1")
E.isogeny_class()
Elliptic curves in class 7488.i
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
7488.i1 | 7488bu3 | \([0, 0, 0, -10956, -430576]\) | \(3044193988/85293\) | \(4074936532992\) | \([2]\) | \(16384\) | \(1.1987\) | |
7488.i2 | 7488bu2 | \([0, 0, 0, -1596, 14960]\) | \(37642192/13689\) | \(163500539904\) | \([2, 2]\) | \(8192\) | \(0.85214\) | |
7488.i3 | 7488bu1 | \([0, 0, 0, -1416, 20504]\) | \(420616192/117\) | \(87340032\) | \([2]\) | \(4096\) | \(0.50557\) | \(\Gamma_0(N)\)-optimal |
7488.i4 | 7488bu4 | \([0, 0, 0, 4884, 105680]\) | \(269676572/257049\) | \(-12280707219456\) | \([2]\) | \(16384\) | \(1.1987\) |
Rank
sage: E.rank()
The elliptic curves in class 7488.i have rank \(1\).
Complex multiplication
The elliptic curves in class 7488.i do not have complex multiplication.Modular form 7488.2.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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.