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SageMath
E = EllipticCurve("be1")
E.isogeny_class()
Elliptic curves in class 185130.be
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 185130.be1 | 185130eq8 | \([1, -1, 0, -219146529465, -39486569687722899]\) | \(901247067798311192691198986281/552431869440\) | \(713448064436549967360\) | \([2]\) | \(637009920\) | \(4.7090\) | |
| 185130.be2 | 185130eq7 | \([1, -1, 0, -13788659385, -608266222557075]\) | \(224494757451893010998773801/6152490825146276160000\) | \(7945744830242795400268319040000\) | \([2]\) | \(637009920\) | \(4.7090\) | |
| 185130.be3 | 185130eq6 | \([1, -1, 0, -13696660665, -616974839792019]\) | \(220031146443748723000125481/172266701724057600\) | \(222476927401897467111014400\) | \([2, 2]\) | \(318504960\) | \(4.3625\) | |
| 185130.be4 | 185130eq5 | \([1, -1, 0, -2706064290, -54141598653444]\) | \(1696892787277117093383481/1440538624914939000\) | \(1860409492184949068088891000\) | \([2]\) | \(212336640\) | \(4.1597\) | |
| 185130.be5 | 185130eq4 | \([1, -1, 0, -1772225010, 28409818632300]\) | \(476646772170172569823801/5862293314453125000\) | \(7570964040499055689453125000\) | \([2]\) | \(212336640\) | \(4.1597\) | |
| 185130.be6 | 185130eq3 | \([1, -1, 0, -850293945, -9775901131155]\) | \(-52643812360427830814761/1504091705903677440\) | \(-1942486260613167613067919360\) | \([2]\) | \(159252480\) | \(4.0159\) | |
| 185130.be7 | 185130eq2 | \([1, -1, 0, -206809290, -441106170444]\) | \(757443433548897303481/373234243041000000\) | \(482020069821412653729000000\) | \([2, 2]\) | \(106168320\) | \(3.8132\) | |
| 185130.be8 | 185130eq1 | \([1, -1, 0, 47232630, -52879308300]\) | \(9023321954633914439/6156756739584000\) | \(-7951254122097610384896000\) | \([2]\) | \(53084160\) | \(3.4666\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 185130.be have rank \(1\).
Complex multiplication
The elliptic curves in class 185130.be do not have complex multiplication.Modular form 185130.2.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 LMFDB numbering.
\(\left(\begin{array}{rrrrrrrr} 1 & 4 & 2 & 3 & 12 & 4 & 6 & 12 \\ 4 & 1 & 2 & 12 & 3 & 4 & 6 & 12 \\ 2 & 2 & 1 & 6 & 6 & 2 & 3 & 6 \\ 3 & 12 & 6 & 1 & 4 & 12 & 2 & 4 \\ 12 & 3 & 6 & 4 & 1 & 12 & 2 & 4 \\ 4 & 4 & 2 & 12 & 12 & 1 & 6 & 3 \\ 6 & 6 & 3 & 2 & 2 & 6 & 1 & 2 \\ 12 & 12 & 6 & 4 & 4 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
