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SageMath
E = EllipticCurve("cr1")
E.isogeny_class()
Elliptic curves in class 85680cr
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 85680.b4 | 85680cr1 | \([0, 0, 0, -13488, 395663]\) | \(628177876549632/206979654175\) | \(89415210603600\) | \([2]\) | \(276480\) | \(1.3800\) | \(\Gamma_0(N)\)-optimal |
| 85680.b3 | 85680cr2 | \([0, 0, 0, -87183, -9612118]\) | \(10602674044119792/361255960625\) | \(2497001199840000\) | \([2]\) | \(552960\) | \(1.7265\) | |
| 85680.b2 | 85680cr3 | \([0, 0, 0, -984528, 376002027]\) | \(335117149277257728/31609375\) | \(9954677250000\) | \([2]\) | \(829440\) | \(1.9293\) | |
| 85680.b1 | 85680cr4 | \([0, 0, 0, -986823, 374160978]\) | \(21091634831728368/203369140625\) | \(1024746187500000000\) | \([2]\) | \(1658880\) | \(2.2758\) |
Rank
sage: E.rank()
The elliptic curves in class 85680cr have rank \(1\).
Complex multiplication
The elliptic curves in class 85680cr do not have complex multiplication.Modular form 85680.2.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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
