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Properties

Label 205350eb
Number of curves $2$
Conductor $205350$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("eb1")
 
E.isogeny_class()
 

Elliptic curves in class 205350eb

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
205350.l2 205350eb1 \([1, 1, 0, -41087825, -11108272875]\) \(306163065625/175056768\) \(4386208718083848750000000\) \([]\) \(62052480\) \(3.4181\) \(\Gamma_0(N)\)-optimal
205350.l1 205350eb2 \([1, 1, 0, -2415447200, -45693307776000]\) \(62202232222815625/232783872\) \(5832614531051520000000000\) \([]\) \(186157440\) \(3.9674\)  

Rank

sage: E.rank()
 

The elliptic curves in class 205350eb have rank \(1\).

Complex multiplication

The elliptic curves in class 205350eb do not have complex multiplication.

Modular form 205350.2.a.eb

sage: E.q_eigenform(10)
 
\(q - q^{2} - q^{3} + q^{4} + q^{6} - 2 q^{7} - q^{8} + q^{9} + 6 q^{11} - q^{12} - q^{13} + 2 q^{14} + q^{16} + 3 q^{17} - q^{18} - 8 q^{19} + O(q^{20})\) 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.

\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.