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SageMath
E = EllipticCurve("bm1")
E.isogeny_class()
Elliptic curves in class 188760bm
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
188760.z4 | 188760bm1 | \([0, -1, 0, -11934996, 17075441220]\) | \(-414566786956390864/37694855859375\) | \(-17095356554519100000000\) | \([2]\) | \(17694720\) | \(3.0090\) | \(\Gamma_0(N)\)-optimal |
188760.z3 | 188760bm2 | \([0, -1, 0, -194947496, 1047728636220]\) | \(451669360369114547716/3285246875625\) | \(5959695605994599040000\) | \([2, 2]\) | \(35389440\) | \(3.3556\) | |
188760.z1 | 188760bm3 | \([0, -1, 0, -3119154496, 67051759357420]\) | \(925014005732729613959858/48938175\) | \(177555378669926400\) | \([2]\) | \(70778880\) | \(3.7022\) | |
188760.z2 | 188760bm4 | \([0, -1, 0, -198940496, 1002574195020]\) | \(239997788713612187858/19220920226046825\) | \(69736514880667114176153600\) | \([2]\) | \(70778880\) | \(3.7022\) |
Rank
sage: E.rank()
The elliptic curves in class 188760bm have rank \(1\).
Complex multiplication
The elliptic curves in class 188760bm do not have complex multiplication.Modular form 188760.2.a.bm
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.