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SageMath
E = EllipticCurve("cr1")
E.isogeny_class()
Elliptic curves in class 160080cr
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 160080.d3 | 160080cr1 | \([0, -1, 0, -150076, 22427776]\) | \(1460227012917743824/450225\) | \(115257600\) | \([2]\) | \(454656\) | \(1.3483\) | \(\Gamma_0(N)\)-optimal |
| 160080.d2 | 160080cr2 | \([0, -1, 0, -150096, 22421520]\) | \(365202721109950276/202702550625\) | \(207567411840000\) | \([2, 2]\) | \(909312\) | \(1.6948\) | |
| 160080.d4 | 160080cr3 | \([0, -1, 0, -123416, 30617616]\) | \(-101510666273096498/138465291796875\) | \(-283576917600000000\) | \([2]\) | \(1818624\) | \(2.0414\) | |
| 160080.d1 | 160080cr4 | \([0, -1, 0, -177096, 13824720]\) | \(299931768520181138/133600200066675\) | \(273613209736550400\) | \([2]\) | \(1818624\) | \(2.0414\) |
Rank
sage: E.rank()
The elliptic curves in class 160080cr have rank \(1\).
Complex multiplication
The elliptic curves in class 160080cr do not have complex multiplication.Modular form 160080.2.a.cr
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.
