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SageMath
E = EllipticCurve("be1")
E.isogeny_class()
Elliptic curves in class 201810be
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
201810.bp7 | 201810be1 | \([1, 1, 1, -39421, 1043603]\) | \(7633736209/3870720\) | \(3435278248120320\) | \([2]\) | \(1451520\) | \(1.6736\) | \(\Gamma_0(N)\)-optimal |
201810.bp5 | 201810be2 | \([1, 1, 1, -346941, -78050541]\) | \(5203798902289/57153600\) | \(50724030382401600\) | \([2, 2]\) | \(2903040\) | \(2.0201\) | |
201810.bp4 | 201810be3 | \([1, 1, 1, -2576461, 1590706739]\) | \(2131200347946769/2058000\) | \(1826482575498000\) | \([2]\) | \(4354560\) | \(2.2229\) | |
201810.bp6 | 201810be4 | \([1, 1, 1, -77861, -195692317]\) | \(-58818484369/18600435000\) | \(-16507954530701235000\) | \([2]\) | \(5806080\) | \(2.3667\) | |
201810.bp2 | 201810be5 | \([1, 1, 1, -5536341, -5016283581]\) | \(21145699168383889/2593080\) | \(2301368045127480\) | \([2]\) | \(5806080\) | \(2.3667\) | |
201810.bp3 | 201810be6 | \([1, 1, 1, -2595681, 1565743803]\) | \(2179252305146449/66177562500\) | \(58732830318357562500\) | \([2, 2]\) | \(8709120\) | \(2.5694\) | |
201810.bp8 | 201810be7 | \([1, 1, 1, 700549, 5274661799]\) | \(42841933504271/13565917968750\) | \(-12039802133409667968750\) | \([2]\) | \(17418240\) | \(2.9160\) | |
201810.bp1 | 201810be8 | \([1, 1, 1, -6199431, -3740417697]\) | \(29689921233686449/10380965400750\) | \(9213145005499265160750\) | \([2]\) | \(17418240\) | \(2.9160\) |
Rank
sage: E.rank()
The elliptic curves in class 201810be have rank \(0\).
Complex multiplication
The elliptic curves in class 201810be do not have complex multiplication.Modular form 201810.2.a.be
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 3 & 4 & 4 & 6 & 12 & 12 \\ 2 & 1 & 6 & 2 & 2 & 3 & 6 & 6 \\ 3 & 6 & 1 & 12 & 12 & 2 & 4 & 4 \\ 4 & 2 & 12 & 1 & 4 & 6 & 3 & 12 \\ 4 & 2 & 12 & 4 & 1 & 6 & 12 & 3 \\ 6 & 3 & 2 & 6 & 6 & 1 & 2 & 2 \\ 12 & 6 & 4 & 3 & 12 & 2 & 1 & 4 \\ 12 & 6 & 4 & 12 & 3 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.