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Properties

Label 19074a
Number of curves $2$
Conductor $19074$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("a1")
 
E.isogeny_class()
 

Elliptic curves in class 19074a

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
19074.d2 19074a1 \([1, 1, 0, -157944, 23574528]\) \(18052771191337/444958272\) \(10740210992520768\) \([2]\) \(193536\) \(1.8598\) \(\Gamma_0(N)\)-optimal
19074.d1 19074a2 \([1, 1, 0, -354464, -47054760]\) \(204055591784617/78708537864\) \(1899832763581412616\) \([2]\) \(387072\) \(2.2064\)  

Rank

sage: E.rank()
 

The elliptic curves in class 19074a have rank \(1\).

Complex multiplication

The elliptic curves in class 19074a do not have complex multiplication.

Modular form 19074.2.a.a

sage: E.q_eigenform(10)
 
\(q - q^{2} - q^{3} + q^{4} + 2 q^{5} + q^{6} + 2 q^{7} - q^{8} + q^{9} - 2 q^{10} - q^{11} - q^{12} - 2 q^{14} - 2 q^{15} + q^{16} - q^{18} + 6 q^{19} + O(q^{20})\) 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 Cremona numbering.

\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.