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SageMath
E = EllipticCurve("j1")
E.isogeny_class()
Elliptic curves in class 21294j
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
21294.v2 | 21294j1 | \([1, -1, 0, -512862, -59250996]\) | \(677353174875/322828856\) | \(7110218217541099752\) | \([3]\) | \(404352\) | \(2.3113\) | \(\Gamma_0(N)\)-optimal |
21294.v1 | 21294j2 | \([1, -1, 0, -34423557, -77729060395]\) | \(280965399667875/175616\) | \(2819695374865893888\) | \([]\) | \(1213056\) | \(2.8606\) |
Rank
sage: E.rank()
The elliptic curves in class 21294j have rank \(0\).
Complex multiplication
The elliptic curves in class 21294j do not have complex multiplication.Modular form 21294.2.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.
\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.