Properties

Label 344850gz
Number of curves $4$
Conductor $344850$
CM no
Rank $1$
Graph

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Show commands: SageMath
sage: E = EllipticCurve("gz1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 344850gz

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
344850.gz3 344850gz1 \([1, 0, 0, -93838, -11005708]\) \(3301293169/22800\) \(631118606250000\) \([2]\) \(1966080\) \(1.6734\) \(\Gamma_0(N)\)-optimal
344850.gz2 344850gz2 \([1, 0, 0, -154338, 4905792]\) \(14688124849/8122500\) \(224836003476562500\) \([2, 2]\) \(3932160\) \(2.0200\)  
344850.gz1 344850gz3 \([1, 0, 0, -1878588, 989452542]\) \(26487576322129/44531250\) \(1232653527832031250\) \([2]\) \(7864320\) \(2.3666\)  
344850.gz4 344850gz4 \([1, 0, 0, 601912, 38937042]\) \(871257511151/527800050\) \(-14609843505907031250\) \([2]\) \(7864320\) \(2.3666\)  

Rank

sage: E.rank()
 

The elliptic curves in class 344850gz have rank \(1\).

Complex multiplication

The elliptic curves in class 344850gz do not have complex multiplication.

Modular form 344850.2.a.gz

sage: E.q_eigenform(10)
 
\(q + q^{2} + q^{3} + q^{4} + q^{6} + q^{8} + q^{9} + q^{12} + 2q^{13} + q^{16} + 2q^{17} + q^{18} + q^{19} + O(q^{20})\)  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.

\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.