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Properties

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

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Show commands: SageMath
E = EllipticCurve("x1")
 
E.isogeny_class()
 

Elliptic curves in class 19074x

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
19074.p2 19074x1 \([1, 1, 1, -3474, 62427]\) \(192100033/38148\) \(920799982212\) \([2]\) \(27648\) \(1.0120\) \(\Gamma_0(N)\)-optimal
19074.p1 19074x2 \([1, 1, 1, -52604, 4621691]\) \(666940371553/37026\) \(893717629794\) \([2]\) \(55296\) \(1.3586\)  

Rank

sage: E.rank()
 

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

Complex multiplication

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

Modular form 19074.2.a.x

sage: E.q_eigenform(10)
 
\(q + q^{2} - q^{3} + q^{4} - 2 q^{5} - q^{6} + 2 q^{7} + q^{8} + q^{9} - 2 q^{10} + q^{11} - q^{12} + 4 q^{13} + 2 q^{14} + 2 q^{15} + q^{16} + q^{18} + 2 q^{19} + O(q^{20})\) 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.