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SageMath
E = EllipticCurve("j1")
E.isogeny_class()
Elliptic curves in class 40656.j
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
40656.j1 | 40656by1 | \([0, -1, 0, -2086322, 1172982219]\) | \(-35431687725461248/440311012911\) | \(-12480605093497985136\) | \([]\) | \(1555200\) | \(2.4747\) | \(\Gamma_0(N)\)-optimal |
40656.j2 | 40656by2 | \([0, -1, 0, 7257298, 5988396471]\) | \(1491325446082364672/1410025768453071\) | \(-39967146566183854621296\) | \([]\) | \(4665600\) | \(3.0240\) |
Rank
sage: E.rank()
The elliptic curves in class 40656.j have rank \(0\).
Complex multiplication
The elliptic curves in class 40656.j do not have complex multiplication.Modular form 40656.2.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 LMFDB 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 LMFDB labels.