Properties

Label 3675.f
Number of curves $4$
Conductor $3675$
CM no
Rank $1$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("f1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 3675.f

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
3675.f1 3675l3 \([1, 0, 0, -137838, 19684167]\) \(157551496201/13125\) \(24127236328125\) \([2]\) \(18432\) \(1.6117\)  
3675.f2 3675l2 \([1, 0, 0, -9213, 261792]\) \(47045881/11025\) \(20266878515625\) \([2, 2]\) \(9216\) \(1.2651\)  
3675.f3 3675l1 \([1, 0, 0, -3088, -62833]\) \(1771561/105\) \(193017890625\) \([2]\) \(4608\) \(0.91856\) \(\Gamma_0(N)\)-optimal
3675.f4 3675l4 \([1, 0, 0, 21412, 1639917]\) \(590589719/972405\) \(-1787538685078125\) \([2]\) \(18432\) \(1.6117\)  

Rank

sage: E.rank()
 

The elliptic curves in class 3675.f have rank \(1\).

Complex multiplication

The elliptic curves in class 3675.f do not have complex multiplication.

Modular form 3675.2.a.f

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

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.