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SageMath
E = EllipticCurve("ii1")
E.isogeny_class()
Elliptic curves in class 124950.ii
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
124950.ii1 | 124950hi3 | \([1, 0, 0, -919388, 256453392]\) | \(46753267515625/11591221248\) | \(21307743571968000000\) | \([2]\) | \(3732480\) | \(2.4191\) | |
124950.ii2 | 124950hi1 | \([1, 0, 0, -313013, -67405983]\) | \(1845026709625/793152\) | \(1458024057000000\) | \([2]\) | \(1244160\) | \(1.8698\) | \(\Gamma_0(N)\)-optimal |
124950.ii3 | 124950hi2 | \([1, 0, 0, -264013, -89210983]\) | \(-1107111813625/1228691592\) | \(-2258661517300125000\) | \([2]\) | \(2488320\) | \(2.2164\) | |
124950.ii4 | 124950hi4 | \([1, 0, 0, 2216612, 1626885392]\) | \(655215969476375/1001033261568\) | \(-1840165034222088000000\) | \([2]\) | \(7464960\) | \(2.7657\) |
Rank
sage: E.rank()
The elliptic curves in class 124950.ii have rank \(0\).
Complex multiplication
The elliptic curves in class 124950.ii do not have complex multiplication.Modular form 124950.2.a.ii
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.
\(\left(\begin{array}{rrrr} 1 & 3 & 6 & 2 \\ 3 & 1 & 2 & 6 \\ 6 & 2 & 1 & 3 \\ 2 & 6 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.