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SageMath
E = EllipticCurve("ba1")
E.isogeny_class()
Elliptic curves in class 22386ba
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
22386.ba3 | 22386ba1 | \([1, 0, 0, -643, 17153]\) | \(-29403487464625/110884842432\) | \(-110884842432\) | \([6]\) | \(20736\) | \(0.80494\) | \(\Gamma_0(N)\)-optimal |
22386.ba2 | 22386ba2 | \([1, 0, 0, -14683, 682649]\) | \(350082141630936625/555332456952\) | \(555332456952\) | \([6]\) | \(41472\) | \(1.1515\) | |
22386.ba4 | 22386ba3 | \([1, 0, 0, 5657, -408475]\) | \(20020616659055375/83832462778428\) | \(-83832462778428\) | \([2]\) | \(62208\) | \(1.3542\) | |
22386.ba1 | 22386ba4 | \([1, 0, 0, -60253, -5035357]\) | \(24191354664255948625/3068177831138238\) | \(3068177831138238\) | \([2]\) | \(124416\) | \(1.7008\) |
Rank
sage: E.rank()
The elliptic curves in class 22386ba have rank \(1\).
Complex multiplication
The elliptic curves in class 22386ba do not have complex multiplication.Modular form 22386.2.a.ba
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.