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SageMath
E = EllipticCurve("f1")
E.isogeny_class()
Elliptic curves in class 130130.f
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
130130.f1 | 130130f4 | \([1, 0, 1, -594884, -175688004]\) | \(4823468134087681/30382271150\) | \(146649419827260350\) | \([2]\) | \(2654208\) | \(2.1313\) | |
130130.f2 | 130130f2 | \([1, 0, 1, -45634, 3587196]\) | \(2177286259681/105875000\) | \(511038402875000\) | \([2]\) | \(884736\) | \(1.5820\) | |
130130.f3 | 130130f3 | \([1, 0, 1, -15214, -5960628]\) | \(-80677568161/3131816380\) | \(-15116679489331420\) | \([2]\) | \(1327104\) | \(1.7847\) | |
130130.f4 | 130130f1 | \([1, 0, 1, 1686, 218012]\) | \(109902239/4312000\) | \(-20813200408000\) | \([2]\) | \(442368\) | \(1.2354\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 130130.f have rank \(2\).
Complex multiplication
The elliptic curves in class 130130.f do not have complex multiplication.Modular form 130130.2.a.f
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.