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SageMath
E = EllipticCurve("cg1")
E.isogeny_class()
Elliptic curves in class 439824.cg
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
439824.cg1 | 439824cg3 | \([0, -1, 0, -1243664312, 16881597321072]\) | \(441453577446719855661097/4354701912\) | \(2098488628203061248\) | \([2]\) | \(99090432\) | \(3.5474\) | \(\Gamma_0(N)\)-optimal* |
439824.cg2 | 439824cg2 | \([0, -1, 0, -77730872, 263781187440]\) | \(107784459654566688937/10704361149504\) | \(5158327848460272009216\) | \([2, 2]\) | \(49545216\) | \(3.2009\) | \(\Gamma_0(N)\)-optimal* |
439824.cg3 | 439824cg4 | \([0, -1, 0, -71866552, 305253658480]\) | \(-85183593440646799657/34223681512621656\) | \(-16492060288116429646823424\) | \([2]\) | \(99090432\) | \(3.5474\) | |
439824.cg4 | 439824cg1 | \([0, -1, 0, -5226552, 3461676912]\) | \(32765849647039657/8229948198912\) | \(3965932239477956149248\) | \([2]\) | \(24772608\) | \(2.8543\) | \(\Gamma_0(N)\)-optimal* |
Rank
sage: E.rank()
The elliptic curves in class 439824.cg have rank \(0\).
Complex multiplication
The elliptic curves in class 439824.cg do not have complex multiplication.Modular form 439824.2.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 LMFDB 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 LMFDB labels.