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SageMath
E = EllipticCurve("c1")
E.isogeny_class()
Elliptic curves in class 4335.c
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
4335.c1 | 4335d7 | \([1, 0, 0, -624246, -189889425]\) | \(1114544804970241/405\) | \(9775715445\) | \([2]\) | \(20480\) | \(1.7075\) | |
4335.c2 | 4335d5 | \([1, 0, 0, -39021, -2968560]\) | \(272223782641/164025\) | \(3959164755225\) | \([2, 2]\) | \(10240\) | \(1.3609\) | |
4335.c3 | 4335d8 | \([1, 0, 0, -31796, -4099995]\) | \(-147281603041/215233605\) | \(-5195215991806245\) | \([2]\) | \(20480\) | \(1.7075\) | |
4335.c4 | 4335d4 | \([1, 0, 0, -23126, 1351701]\) | \(56667352321/15\) | \(362063535\) | \([2]\) | \(5120\) | \(1.0143\) | |
4335.c5 | 4335d3 | \([1, 0, 0, -2896, -27985]\) | \(111284641/50625\) | \(1221964430625\) | \([2, 2]\) | \(5120\) | \(1.0143\) | |
4335.c6 | 4335d2 | \([1, 0, 0, -1451, 20856]\) | \(13997521/225\) | \(5430953025\) | \([2, 2]\) | \(2560\) | \(0.66776\) | |
4335.c7 | 4335d1 | \([1, 0, 0, -6, 915]\) | \(-1/15\) | \(-362063535\) | \([2]\) | \(1280\) | \(0.32118\) | \(\Gamma_0(N)\)-optimal |
4335.c8 | 4335d6 | \([1, 0, 0, 10109, -207454]\) | \(4733169839/3515625\) | \(-84858641015625\) | \([2]\) | \(10240\) | \(1.3609\) |
Rank
sage: E.rank()
The elliptic curves in class 4335.c have rank \(0\).
Complex multiplication
The elliptic curves in class 4335.c do not have complex multiplication.Modular form 4335.2.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 LMFDB numbering.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 16 & 4 & 8 & 16 & 8 \\ 2 & 1 & 2 & 8 & 2 & 4 & 8 & 4 \\ 4 & 2 & 1 & 16 & 4 & 8 & 16 & 8 \\ 16 & 8 & 16 & 1 & 4 & 2 & 4 & 8 \\ 4 & 2 & 4 & 4 & 1 & 2 & 4 & 2 \\ 8 & 4 & 8 & 2 & 2 & 1 & 2 & 4 \\ 16 & 8 & 16 & 4 & 4 & 2 & 1 & 8 \\ 8 & 4 & 8 & 8 & 2 & 4 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.