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SageMath
E = EllipticCurve("r1")
E.isogeny_class()
Elliptic curves in class 316239r
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
316239.r2 | 316239r1 | \([1, 0, 1, 5088544, 3201402449]\) | \(5679290619623375/5011824295863\) | \(-12858969953163528545967\) | \([2]\) | \(15759360\) | \(2.9298\) | \(\Gamma_0(N)\)-optimal |
316239.r1 | 316239r2 | \([1, 0, 1, -25556521, 28465194035]\) | \(719479696477074625/279917719389693\) | \(718192284985186692502437\) | \([2]\) | \(31518720\) | \(3.2764\) |
Rank
sage: E.rank()
The elliptic curves in class 316239r have rank \(1\).
Complex multiplication
The elliptic curves in class 316239r do not have complex multiplication.Modular form 316239.2.a.r
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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.