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SageMath
E = EllipticCurve("x1")
E.isogeny_class()
Elliptic curves in class 42978x
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
42978.v2 | 42978x1 | \([1, 0, 0, -1493, 25569]\) | \(-368062283004625/72186536448\) | \(-72186536448\) | \([3]\) | \(38880\) | \(0.80676\) | \(\Gamma_0(N)\)-optimal |
42978.v3 | 42978x2 | \([1, 0, 0, 10387, -132327]\) | \(123934293043859375/79385028425352\) | \(-79385028425352\) | \([3]\) | \(116640\) | \(1.3561\) | |
42978.v1 | 42978x3 | \([1, 0, 0, -174863, -29029845]\) | \(-591313293127508640625/21499590336237858\) | \(-21499590336237858\) | \([]\) | \(349920\) | \(1.9054\) |
Rank
sage: E.rank()
The elliptic curves in class 42978x have rank \(1\).
Complex multiplication
The elliptic curves in class 42978x do not have complex multiplication.Modular form 42978.2.a.x
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.
\(\left(\begin{array}{rrr} 1 & 3 & 9 \\ 3 & 1 & 3 \\ 9 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.