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SageMath
E = EllipticCurve("bf1")
E.isogeny_class()
Elliptic curves in class 232050bf
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 232050.bf7 | 232050bf1 | \([1, 1, 0, -7450250, 36654796500]\) | \(-2926956820564562516641/35459588343029760000\) | \(-554056067859840000000000\) | \([2]\) | \(37748736\) | \(3.2378\) | \(\Gamma_0(N)\)-optimal |
| 232050.bf6 | 232050bf2 | \([1, 1, 0, -217402250, 1229812012500]\) | \(72727020009972527154752161/265361167808100000000\) | \(4146268247001562500000000\) | \([2, 2]\) | \(75497472\) | \(3.5844\) | |
| 232050.bf3 | 232050bf3 | \([1, 1, 0, -3475384250, 78857749126500]\) | \(297106512928238351998640242081/3977028808593750000\) | \(62141075134277343750000\) | \([2]\) | \(150994944\) | \(3.9309\) | |
| 232050.bf5 | 232050bf4 | \([1, 1, 0, -318652250, -31864237500]\) | \(229010110533436633465952161/132501160769452503210000\) | \(2070330637022695362656250000\) | \([2, 2]\) | \(150994944\) | \(3.9309\) | |
| 232050.bf8 | 232050bf5 | \([1, 1, 0, 1274460250, -253306875000]\) | \(14651516183052242700771495839/8480668142378708755560900\) | \(-132510439724667324305639062500\) | \([2]\) | \(301989888\) | \(4.2775\) | |
| 232050.bf2 | 232050bf6 | \([1, 1, 0, -3531764750, -80555676600000]\) | \(311802066473807207098058600161/1033693082103011001480900\) | \(16151454407859546898139062500\) | \([2, 2]\) | \(301989888\) | \(4.2775\) | |
| 232050.bf4 | 232050bf7 | \([1, 1, 0, -2010131000, -150427576766250]\) | \(-57487943130312093140621093761/592356094985924086700006670\) | \(-9255563984155063854687604218750\) | \([2]\) | \(603979776\) | \(4.6241\) | |
| 232050.bf1 | 232050bf8 | \([1, 1, 0, -56463198500, -5164143505383750]\) | \(1274090022584975661628188489514561/14072533302105480763470\) | \(219883332845398136929218750\) | \([2]\) | \(603979776\) | \(4.6241\) |
Rank
sage: E.rank()
The elliptic curves in class 232050bf have rank \(0\).
Complex multiplication
The elliptic curves in class 232050bf do not have complex multiplication.Modular form 232050.2.a.bf
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 4 & 8 & 8 & 16 & 16 \\ 2 & 1 & 2 & 2 & 4 & 4 & 8 & 8 \\ 4 & 2 & 1 & 4 & 8 & 8 & 16 & 16 \\ 4 & 2 & 4 & 1 & 2 & 2 & 4 & 4 \\ 8 & 4 & 8 & 2 & 1 & 4 & 8 & 8 \\ 8 & 4 & 8 & 2 & 4 & 1 & 2 & 2 \\ 16 & 8 & 16 & 4 & 8 & 2 & 1 & 4 \\ 16 & 8 & 16 & 4 & 8 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
