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SageMath
E = EllipticCurve("c1")
E.isogeny_class()
Elliptic curves in class 8470.c
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
8470.c1 | 8470f4 | \([1, 0, 1, -3160039, 2100623886]\) | \(1969902499564819009/63690429687500\) | \(112831481307617187500\) | \([2]\) | \(414720\) | \(2.6216\) | |
8470.c2 | 8470f2 | \([1, 0, 1, -432699, -108616378]\) | \(5057359576472449/51765560000\) | \(91705847239160000\) | \([2]\) | \(138240\) | \(2.0723\) | |
8470.c3 | 8470f1 | \([1, 0, 1, -6779, -4180794]\) | \(-19443408769/4249907200\) | \(-7528969849139200\) | \([2]\) | \(69120\) | \(1.7257\) | \(\Gamma_0(N)\)-optimal |
8470.c4 | 8470f3 | \([1, 0, 1, 60981, 112610342]\) | \(14156681599871/3100231750000\) | \(-5492249659261750000\) | \([2]\) | \(207360\) | \(2.2750\) |
Rank
sage: E.rank()
The elliptic curves in class 8470.c have rank \(0\).
Complex multiplication
The elliptic curves in class 8470.c do not have complex multiplication.Modular form 8470.2.a.c
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.
\(\left(\begin{array}{rrrr} 1 & 3 & 6 & 2 \\ 3 & 1 & 2 & 6 \\ 6 & 2 & 1 & 3 \\ 2 & 6 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.