Properties

Label 3675.i
Number of curves $2$
Conductor $3675$
CM no
Rank $1$
Graph

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

Elliptic curves in class 3675.i

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
3675.i1 3675i2 \([0, 1, 1, -5133, 140144]\) \(-19539165184/46875\) \(-35888671875\) \([]\) \(3456\) \(0.90464\)  
3675.i2 3675i1 \([0, 1, 1, 117, 1019]\) \(229376/675\) \(-516796875\) \([]\) \(1152\) \(0.35533\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

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

Complex multiplication

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

Modular form 3675.2.a.i

sage: E.q_eigenform(10)
 
\(q + q^{3} - 2q^{4} + q^{9} - 2q^{12} - q^{13} + 4q^{16} + 6q^{17} - 5q^{19} + O(q^{20})\)  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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.