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SageMath
E = EllipticCurve("c1")
E.isogeny_class()
Elliptic curves in class 3192.c
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
3192.c1 | 3192b3 | \([0, -1, 0, -20104, -1089956]\) | \(877592260337188/494771571\) | \(506646088704\) | \([2]\) | \(6144\) | \(1.1928\) | |
3192.c2 | 3192b4 | \([0, -1, 0, -11744, 486588]\) | \(174947951977348/2957342913\) | \(3028319142912\) | \([2]\) | \(6144\) | \(1.1928\) | |
3192.c3 | 3192b2 | \([0, -1, 0, -1484, -9996]\) | \(1412791482832/631868769\) | \(161758404864\) | \([2, 2]\) | \(3072\) | \(0.84622\) | |
3192.c4 | 3192b1 | \([0, -1, 0, 321, -1332]\) | \(227910944768/172414683\) | \(-2758634928\) | \([4]\) | \(1536\) | \(0.49965\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 3192.c have rank \(0\).
Complex multiplication
The elliptic curves in class 3192.c do not have complex multiplication.Modular form 3192.2.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.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.