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SageMath
E = EllipticCurve("f1")
E.isogeny_class()
Elliptic curves in class 770f
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
770.f3 | 770f1 | \([1, 0, 0, -56, 3136]\) | \(-19443408769/4249907200\) | \(-4249907200\) | \([6]\) | \(576\) | \(0.52678\) | \(\Gamma_0(N)\)-optimal |
770.f2 | 770f2 | \([1, 0, 0, -3576, 81280]\) | \(5057359576472449/51765560000\) | \(51765560000\) | \([6]\) | \(1152\) | \(0.87336\) | |
770.f4 | 770f3 | \([1, 0, 0, 504, -84560]\) | \(14156681599871/3100231750000\) | \(-3100231750000\) | \([2]\) | \(1728\) | \(1.0761\) | |
770.f1 | 770f4 | \([1, 0, 0, -26116, -1580604]\) | \(1969902499564819009/63690429687500\) | \(63690429687500\) | \([2]\) | \(3456\) | \(1.4227\) |
Rank
sage: E.rank()
The elliptic curves in class 770f have rank \(1\).
Complex multiplication
The elliptic curves in class 770f do not have complex multiplication.Modular form 770.2.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 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.