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SageMath
E = EllipticCurve("j1")
E.isogeny_class()
Elliptic curves in class 10005j
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
10005.m1 | 10005j1 | \([1, 0, 1, -374, 2747]\) | \(5763259856089/450225\) | \(450225\) | \([2]\) | \(1728\) | \(0.13203\) | \(\Gamma_0(N)\)-optimal |
10005.m2 | 10005j2 | \([1, 0, 1, -349, 3137]\) | \(-4681768588489/1621620405\) | \(-1621620405\) | \([2]\) | \(3456\) | \(0.47860\) |
Rank
sage: E.rank()
The elliptic curves in class 10005j have rank \(1\).
Complex multiplication
The elliptic curves in class 10005j do not have complex multiplication.Modular form 10005.2.a.j
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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.