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SageMath
E = EllipticCurve("dt1")
E.isogeny_class()
Elliptic curves in class 428400.dt
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
428400.dt1 | 428400dt3 | \([0, 0, 0, -40807670475, 3172932396252250]\) | \(322159999717985454060440834/4250799\) | \(99162639072000000\) | \([2]\) | \(283115520\) | \(4.2450\) | \(\Gamma_0(N)\)-optimal* |
428400.dt2 | 428400dt4 | \([0, 0, 0, -2557040475, 49309173630250]\) | \(79260902459030376659234/842751810121431609\) | \(19659714226512756574752000000\) | \([2]\) | \(283115520\) | \(4.2450\) | |
428400.dt3 | 428400dt2 | \([0, 0, 0, -2550479475, 49577065821250]\) | \(157304700372188331121828/18069292138401\) | \(210760223502309264000000\) | \([2, 2]\) | \(141557760\) | \(3.8985\) | \(\Gamma_0(N)\)-optimal* |
428400.dt4 | 428400dt1 | \([0, 0, 0, -158994975, 778824598750]\) | \(-152435594466395827792/1646846627220711\) | \(-4802204764975593276000000\) | \([2]\) | \(70778880\) | \(3.5519\) | \(\Gamma_0(N)\)-optimal* |
Rank
sage: E.rank()
The elliptic curves in class 428400.dt have rank \(1\).
Complex multiplication
The elliptic curves in class 428400.dt do not have complex multiplication.Modular form 428400.2.a.dt
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.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.