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SageMath
E = EllipticCurve("d1")
E.isogeny_class()
Elliptic curves in class 19110.d
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 19110.d1 | 19110m5 | \([1, 1, 0, -41761402, 103855143316]\) | \(68463752473882049153689/1817088000000000\) | \(213778586112000000000\) | \([2]\) | \(2239488\) | \(3.0062\) | |
| 19110.d2 | 19110m6 | \([1, 1, 0, -40130682, 112340431764]\) | \(-60752633741424905775769/11197265625000000000\) | \(-1317347103515625000000000\) | \([2]\) | \(4478976\) | \(3.3528\) | |
| 19110.d3 | 19110m3 | \([1, 1, 0, -894667, -93657731]\) | \(673163386034885929/357608625192000\) | \(42072297145213608000\) | \([2]\) | \(746496\) | \(2.4569\) | |
| 19110.d4 | 19110m1 | \([1, 1, 0, -702832, -227083604]\) | \(326355561310674169/465699780\) | \(54789113417220\) | \([2]\) | \(248832\) | \(1.9076\) | \(\Gamma_0(N)\)-optimal |
| 19110.d5 | 19110m2 | \([1, 1, 0, -696462, -231393546]\) | \(-317562142497484249/12339342574650\) | \(-1451711314564997850\) | \([2]\) | \(497664\) | \(2.2542\) | |
| 19110.d6 | 19110m4 | \([1, 1, 0, 3411453, -728379819]\) | \(37321015309599759191/23553520979625000\) | \(-2771048189731901625000\) | \([2]\) | \(1492992\) | \(2.8035\) |
Rank
sage: E.rank()
The elliptic curves in class 19110.d have rank \(1\).
Complex multiplication
The elliptic curves in class 19110.d do not have complex multiplication.Modular form 19110.2.a.d
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 3 & 9 & 18 & 6 \\ 2 & 1 & 6 & 18 & 9 & 3 \\ 3 & 6 & 1 & 3 & 6 & 2 \\ 9 & 18 & 3 & 1 & 2 & 6 \\ 18 & 9 & 6 & 2 & 1 & 3 \\ 6 & 3 & 2 & 6 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
