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SageMath
E = EllipticCurve("c1")
E.isogeny_class()
Elliptic curves in class 225c
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
225.b7 | 225c1 | \([1, -1, 1, -5, -628]\) | \(-1/15\) | \(-170859375\) | \([4]\) | \(48\) | \(0.25860\) | \(\Gamma_0(N)\)-optimal |
225.b6 | 225c2 | \([1, -1, 1, -1130, -14128]\) | \(13997521/225\) | \(2562890625\) | \([2, 2]\) | \(96\) | \(0.60517\) | |
225.b4 | 225c3 | \([1, -1, 1, -18005, -925378]\) | \(56667352321/15\) | \(170859375\) | \([2]\) | \(192\) | \(0.95175\) | |
225.b5 | 225c4 | \([1, -1, 1, -2255, 19622]\) | \(111284641/50625\) | \(576650390625\) | \([2, 2]\) | \(192\) | \(0.95175\) | |
225.b2 | 225c5 | \([1, -1, 1, -30380, 2044622]\) | \(272223782641/164025\) | \(1868347265625\) | \([2, 2]\) | \(384\) | \(1.2983\) | |
225.b8 | 225c6 | \([1, -1, 1, 7870, 141122]\) | \(4733169839/3515625\) | \(-40045166015625\) | \([2]\) | \(384\) | \(1.2983\) | |
225.b1 | 225c7 | \([1, -1, 1, -486005, 130530872]\) | \(1114544804970241/405\) | \(4613203125\) | \([2]\) | \(768\) | \(1.6449\) | |
225.b3 | 225c8 | \([1, -1, 1, -24755, 2820872]\) | \(-147281603041/215233605\) | \(-2451645281953125\) | \([2]\) | \(768\) | \(1.6449\) |
Rank
sage: E.rank()
The elliptic curves in class 225c have rank \(0\).
Complex multiplication
The elliptic curves in class 225c do not have complex multiplication.Modular form 225.2.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 Cremona numbering.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 4 & 8 & 8 & 16 & 16 \\ 2 & 1 & 2 & 2 & 4 & 4 & 8 & 8 \\ 4 & 2 & 1 & 4 & 8 & 8 & 16 & 16 \\ 4 & 2 & 4 & 1 & 2 & 2 & 4 & 4 \\ 8 & 4 & 8 & 2 & 1 & 4 & 2 & 2 \\ 8 & 4 & 8 & 2 & 4 & 1 & 8 & 8 \\ 16 & 8 & 16 & 4 & 2 & 8 & 1 & 4 \\ 16 & 8 & 16 & 4 & 2 & 8 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.