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SageMath
E = EllipticCurve("cc1")
E.isogeny_class()
Elliptic curves in class 215985cc
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
215985.bh4 | 215985cc1 | \([1, 1, 0, -21903, 2826432]\) | \(-656008386769/1581036975\) | \(-2800903444467975\) | \([2]\) | \(1105920\) | \(1.6522\) | \(\Gamma_0(N)\)-optimal |
215985.bh3 | 215985cc2 | \([1, 1, 0, -462948, 120938283]\) | \(6193921595708449/6452105625\) | \(11430298693130625\) | \([2, 2]\) | \(2211840\) | \(1.9988\) | |
215985.bh1 | 215985cc3 | \([1, 1, 0, -7405323, 7753385358]\) | \(25351269426118370449/27551475\) | \(48809118602475\) | \([2]\) | \(4423680\) | \(2.3453\) | |
215985.bh2 | 215985cc4 | \([1, 1, 0, -577293, 56470572]\) | \(12010404962647729/6166198828125\) | \(10923797362151953125\) | \([2]\) | \(4423680\) | \(2.3453\) |
Rank
sage: E.rank()
The elliptic curves in class 215985cc have rank \(1\).
Complex multiplication
The elliptic curves in class 215985cc do not have complex multiplication.Modular form 215985.2.a.cc
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.