Properties

Label 98315.e
Number of curves $3$
Conductor $98315$
CM no
Rank $1$
Graph

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Show commands: SageMath
sage: E = EllipticCurve("e1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 98315.e

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
98315.e1 98315i3 \([0, -1, 1, -368915, -90097869]\) \(-250523582464/13671875\) \(-303028374810546875\) \([]\) \(864864\) \(2.1126\)  
98315.e2 98315i1 \([0, -1, 1, -3745, 99121]\) \(-262144/35\) \(-775752639515\) \([]\) \(96096\) \(1.0140\) \(\Gamma_0(N)\)-optimal
98315.e3 98315i2 \([0, -1, 1, 24345, -257622]\) \(71991296/42875\) \(-950296983405875\) \([]\) \(288288\) \(1.5633\)  

Rank

sage: E.rank()
 

The elliptic curves in class 98315.e have rank \(1\).

Complex multiplication

The elliptic curves in class 98315.e do not have complex multiplication.

Modular form 98315.2.a.e

sage: E.q_eigenform(10)
 
\(q - q^{3} - 2 q^{4} + q^{5} + q^{7} - 2 q^{9} - 3 q^{11} + 2 q^{12} + 5 q^{13} - q^{15} + 4 q^{16} + 3 q^{17} - 2 q^{19} + O(q^{20})\)  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 LMFDB numbering.

\(\left(\begin{array}{rrr} 1 & 9 & 3 \\ 9 & 1 & 3 \\ 3 & 3 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.