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SageMath
E = EllipticCurve("o1")
E.isogeny_class()
Elliptic curves in class 30030.o
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
30030.o1 | 30030o4 | \([1, 0, 1, -4180534, -3245570104]\) | \(8080139196092838808461529/126414009350620992000\) | \(126414009350620992000\) | \([2]\) | \(1492992\) | \(2.6580\) | |
30030.o2 | 30030o2 | \([1, 0, 1, -437119, 109079402]\) | \(9236795664301877985769/201861773047214280\) | \(201861773047214280\) | \([6]\) | \(497664\) | \(2.1087\) | |
30030.o3 | 30030o3 | \([1, 0, 1, -20534, -140546104]\) | \(-957445322254221529/8532623683584000000\) | \(-8532623683584000000\) | \([2]\) | \(746496\) | \(2.3115\) | |
30030.o4 | 30030o1 | \([1, 0, 1, 2281, 5205242]\) | \(1313328092999831/11704749246590400\) | \(-11704749246590400\) | \([6]\) | \(248832\) | \(1.7621\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 30030.o have rank \(0\).
Complex multiplication
The elliptic curves in class 30030.o do not have complex multiplication.Modular form 30030.2.a.o
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.
\(\left(\begin{array}{rrrr} 1 & 3 & 2 & 6 \\ 3 & 1 & 6 & 2 \\ 2 & 6 & 1 & 3 \\ 6 & 2 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.