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SageMath
E = EllipticCurve("z1")
E.isogeny_class()
Elliptic curves in class 214200z
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 214200.dr4 | 214200z1 | \([0, 0, 0, -158994975, -778824598750]\) | \(-152435594466395827792/1646846627220711\) | \(-4802204764975593276000000\) | \([2]\) | \(35389440\) | \(3.5519\) | \(\Gamma_0(N)\)-optimal |
| 214200.dr3 | 214200z2 | \([0, 0, 0, -2550479475, -49577065821250]\) | \(157304700372188331121828/18069292138401\) | \(210760223502309264000000\) | \([2, 2]\) | \(70778880\) | \(3.8985\) | |
| 214200.dr2 | 214200z3 | \([0, 0, 0, -2557040475, -49309173630250]\) | \(79260902459030376659234/842751810121431609\) | \(19659714226512756574752000000\) | \([2]\) | \(141557760\) | \(4.2450\) | |
| 214200.dr1 | 214200z4 | \([0, 0, 0, -40807670475, -3172932396252250]\) | \(322159999717985454060440834/4250799\) | \(99162639072000000\) | \([2]\) | \(141557760\) | \(4.2450\) |
Rank
sage: E.rank()
The elliptic curves in class 214200z have rank \(1\).
Complex multiplication
The elliptic curves in class 214200z do not have complex multiplication.Modular form 214200.2.a.z
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
