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SageMath
E = EllipticCurve("cg1")
E.isogeny_class()
Elliptic curves in class 193200cg
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
193200.eq4 | 193200cg1 | \([0, 1, 0, -456408, -355480812]\) | \(-164287467238609/757170892800\) | \(-48458937139200000000\) | \([2]\) | \(5308416\) | \(2.4620\) | \(\Gamma_0(N)\)-optimal |
193200.eq3 | 193200cg2 | \([0, 1, 0, -10824408, -13688728812]\) | \(2191574502231419089/4115217960000\) | \(263373949440000000000\) | \([2, 2]\) | \(10616832\) | \(2.8086\) | |
193200.eq2 | 193200cg3 | \([0, 1, 0, -14424408, -3803128812]\) | \(5186062692284555089/2903809817953800\) | \(185843828349043200000000\) | \([2]\) | \(21233664\) | \(3.1551\) | |
193200.eq1 | 193200cg4 | \([0, 1, 0, -173112408, -876736312812]\) | \(8964546681033941529169/31696875000\) | \(2028600000000000000\) | \([2]\) | \(21233664\) | \(3.1551\) |
Rank
sage: E.rank()
The elliptic curves in class 193200cg have rank \(1\).
Complex multiplication
The elliptic curves in class 193200cg do not have complex multiplication.Modular form 193200.2.a.cg
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.