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SageMath
E = EllipticCurve("i1")
E.isogeny_class()
Elliptic curves in class 4725.i
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
4725.i1 | 4725c3 | \([0, 0, 1, -95850, -11421844]\) | \(35184082944/7\) | \(19375453125\) | \([]\) | \(11664\) | \(1.3634\) | |
4725.i2 | 4725c2 | \([0, 0, 1, -1350, -10969]\) | \(884736/343\) | \(105488578125\) | \([]\) | \(3888\) | \(0.81405\) | |
4725.i3 | 4725c1 | \([0, 0, 1, -600, 5656]\) | \(56623104/7\) | \(2953125\) | \([]\) | \(1296\) | \(0.26474\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 4725.i have rank \(1\).
Complex multiplication
The elliptic curves in class 4725.i do not have complex multiplication.Modular form 4725.2.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.
\(\left(\begin{array}{rrr} 1 & 3 & 9 \\ 3 & 1 & 3 \\ 9 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.