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Properties

Label 75810.i
Number of curves $4$
Conductor $75810$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("i1")
 
E.isogeny_class()
 

Elliptic curves in class 75810.i

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
75810.i1 75810o4 \([1, 1, 0, -53561938, -150765888812]\) \(361219316414914078129/378697617819360\) \(17816163062913090056160\) \([2]\) \(11059200\) \(3.1867\)  
75810.i2 75810o2 \([1, 1, 0, -4177138, -1100313932]\) \(171332100266282929/88068464870400\) \(4143258518145518822400\) \([2, 2]\) \(5529600\) \(2.8402\)  
75810.i3 75810o1 \([1, 1, 0, -2328818, 1354624692]\) \(29689921233686449/307510640640\) \(14467109005783203840\) \([2]\) \(2764800\) \(2.4936\) \(\Gamma_0(N)\)-optimal
75810.i4 75810o3 \([1, 1, 0, 15634542, -8513844588]\) \(8983747840943130191/5865547515660000\) \(-275949850421585996460000\) \([2]\) \(11059200\) \(3.1867\)  

Rank

sage: E.rank()
 

The elliptic curves in class 75810.i have rank \(1\).

Complex multiplication

The elliptic curves in class 75810.i do not have complex multiplication.

Modular form 75810.2.a.i

sage: E.q_eigenform(10)
 
\(q - q^{2} - q^{3} + q^{4} - q^{5} + q^{6} + q^{7} - q^{8} + q^{9} + q^{10} - 4 q^{11} - q^{12} + 6 q^{13} - q^{14} + q^{15} + q^{16} - 2 q^{17} - q^{18} + O(q^{20})\) 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 LMFDB 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 LMFDB labels.