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SageMath
E = EllipticCurve("dw1")
E.isogeny_class()
Elliptic curves in class 433200dw
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
433200.dw3 | 433200dw1 | \([0, -1, 0, -3735775408, 88693982053312]\) | \(-1914980734749238129/20440940544000\) | \(-61546371607110352896000000000\) | \([2]\) | \(597196800\) | \(4.3406\) | \(\Gamma_0(N)\)-optimal* |
433200.dw2 | 433200dw2 | \([0, -1, 0, -59924703408, 5646228472677312]\) | \(7903870428425797297009/886464000000\) | \(2669086710706176000000000000\) | \([2]\) | \(1194393600\) | \(4.6871\) | \(\Gamma_0(N)\)-optimal* |
433200.dw4 | 433200dw3 | \([0, -1, 0, 12344608592, 461681261413312]\) | \(69096190760262356111/70568821500000000\) | \(-212478232230351456000000000000000\) | \([2]\) | \(1791590400\) | \(4.8899\) | |
433200.dw1 | 433200dw4 | \([0, -1, 0, -66890559408, 4251974757861312]\) | \(10993009831928446009969/3767761230468750000\) | \(11344489375042968750000000000000000\) | \([2]\) | \(3583180800\) | \(5.2364\) |
Rank
sage: E.rank()
The elliptic curves in class 433200dw have rank \(1\).
Complex multiplication
The elliptic curves in class 433200dw do not have complex multiplication.Modular form 433200.2.a.dw
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.