Properties

Label 1849.b
Number of curves $2$
Conductor $1849$
CM \(\Q(\sqrt{-43}) \)
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("b1")
 
E.isogeny_class()
 

Elliptic curves in class 1849.b

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality CM discriminant
1849.b1 1849a2 \([0, 0, 1, -1590140, -771794326]\) \(-884736000\) \(-502592611936843\) \([]\) \(11352\) \(2.1060\)   \(-43\)
1849.b2 1849a1 \([0, 0, 1, -860, 9707]\) \(-884736000\) \(-79507\) \([]\) \(264\) \(0.22543\) \(\Gamma_0(N)\)-optimal \(-43\)

Rank

sage: E.rank()
 

The elliptic curves in class 1849.b have rank \(1\).

Complex multiplication

Each elliptic curve in class 1849.b has complex multiplication by an order in the imaginary quadratic field \(\Q(\sqrt{-43}) \).

Modular form 1849.2.a.b

sage: E.q_eigenform(10)
 
\(q - 2 q^{4} - 3 q^{9} - q^{11} + 3 q^{13} + 4 q^{16} + 5 q^{17} + O(q^{20})\) Copy content Toggle raw display

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.

\(\left(\begin{array}{rr} 1 & 43 \\ 43 & 1 \end{array}\right)\)

Isogeny graph

sage: E.isogeny_graph().plot(edge_labels=True)
 

The vertices are labelled with LMFDB labels.