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SageMath
E = EllipticCurve("i1")
E.isogeny_class()
Elliptic curves in class 8470i
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
8470.m1 | 8470i1 | \([1, 0, 1, -4964, 227962]\) | \(-63088729/68600\) | \(-14705019236600\) | \([3]\) | \(19008\) | \(1.2214\) | \(\Gamma_0(N)\)-optimal |
8470.m2 | 8470i2 | \([1, 0, 1, 41621, -4132394]\) | \(37199299511/56000000\) | \(-12004097336000000\) | \([]\) | \(57024\) | \(1.7707\) |
Rank
sage: E.rank()
The elliptic curves in class 8470i have rank \(1\).
Complex multiplication
The elliptic curves in class 8470i do not have complex multiplication.Modular form 8470.2.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 Cremona numbering.
\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.