Properties

Label 3675.j
Number of curves $8$
Conductor $3675$
CM no
Rank $0$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("j1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 3675.j

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
3675.j1 3675e7 \([1, 1, 0, -2646025, 1655579500]\) \(1114544804970241/405\) \(744497578125\) \([2]\) \(36864\) \(2.0685\)  
3675.j2 3675e5 \([1, 1, 0, -165400, 25808875]\) \(272223782641/164025\) \(301521519140625\) \([2, 2]\) \(18432\) \(1.7220\)  
3675.j3 3675e8 \([1, 1, 0, -134775, 35700750]\) \(-147281603041/215233605\) \(-395656537416328125\) \([2]\) \(36864\) \(2.0685\)  
3675.j4 3675e3 \([1, 1, 0, -98025, -11853750]\) \(56667352321/15\) \(27573984375\) \([2]\) \(9216\) \(1.3754\)  
3675.j5 3675e4 \([1, 1, 0, -12275, 237000]\) \(111284641/50625\) \(93062197265625\) \([2, 2]\) \(9216\) \(1.3754\)  
3675.j6 3675e2 \([1, 1, 0, -6150, -185625]\) \(13997521/225\) \(413609765625\) \([2, 2]\) \(4608\) \(1.0288\)  
3675.j7 3675e1 \([1, 1, 0, -25, -8000]\) \(-1/15\) \(-27573984375\) \([2]\) \(2304\) \(0.68225\) \(\Gamma_0(N)\)-optimal
3675.j8 3675e6 \([1, 1, 0, 42850, 1835625]\) \(4733169839/3515625\) \(-6462652587890625\) \([2]\) \(18432\) \(1.7220\)  

Rank

sage: E.rank()
 

The elliptic curves in class 3675.j have rank \(0\).

Complex multiplication

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

Modular form 3675.2.a.j

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

\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 16 & 4 & 8 & 16 & 8 \\ 2 & 1 & 2 & 8 & 2 & 4 & 8 & 4 \\ 4 & 2 & 1 & 16 & 4 & 8 & 16 & 8 \\ 16 & 8 & 16 & 1 & 4 & 2 & 4 & 8 \\ 4 & 2 & 4 & 4 & 1 & 2 & 4 & 2 \\ 8 & 4 & 8 & 2 & 2 & 1 & 2 & 4 \\ 16 & 8 & 16 & 4 & 4 & 2 & 1 & 8 \\ 8 & 4 & 8 & 8 & 2 & 4 & 8 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.