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SageMath
E = EllipticCurve("o1")
E.isogeny_class()
Elliptic curves in class 209814.o
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
209814.o1 | 209814cw3 | \([1, 1, 0, -580295449120, 170145804044320768]\) | \(505384091400037554067434625/815656731648\) | \(34878440955798278848914432\) | \([2]\) | \(1194393600\) | \(5.0565\) | |
209814.o2 | 209814cw4 | \([1, 1, 0, -580289854080, 170149249090609056]\) | \(-505369473241574671219626625/20303219722982711328\) | \(-868189549407475871603066198667552\) | \([2]\) | \(2388787200\) | \(5.4031\) | |
209814.o3 | 209814cw1 | \([1, 1, 0, -7184311840, 232013155972096]\) | \(959024269496848362625/11151660319506432\) | \(476858108223003928952504844288\) | \([2]\) | \(398131200\) | \(4.5072\) | \(\Gamma_0(N)\)-optimal |
209814.o4 | 209814cw2 | \([1, 1, 0, -1454990880, 591875243062272]\) | \(-7966267523043306625/3534510366354604032\) | \(-151139819408438618294904240242688\) | \([2]\) | \(796262400\) | \(4.8538\) |
Rank
sage: E.rank()
The elliptic curves in class 209814.o have rank \(0\).
Complex multiplication
The elliptic curves in class 209814.o do not have complex multiplication.Modular form 209814.2.a.o
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.