↑

Properties

Label 975a
Number of curves $8$
Conductor $975$
CM no
Rank $1$
Graph

Related objects

Downloads

Learn more

Show commands: SageMath
E = EllipticCurve("a1")
 
E.isogeny_class()
 

Elliptic curves in class 975a

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
975.i6 975a1 \([1, 1, 0, -2750, 54375]\) \(147281603041/5265\) \(82265625\) \([2]\) \(576\) \(0.60784\) \(\Gamma_0(N)\)-optimal
975.i5 975a2 \([1, 1, 0, -2875, 49000]\) \(168288035761/27720225\) \(433128515625\) \([2, 2]\) \(1152\) \(0.95442\)  
975.i4 975a3 \([1, 1, 0, -13000, -528125]\) \(15551989015681/1445900625\) \(22592197265625\) \([2, 2]\) \(2304\) \(1.3010\)  
975.i7 975a4 \([1, 1, 0, 5250, 284625]\) \(1023887723039/2798036865\) \(-43719326015625\) \([2]\) \(2304\) \(1.3010\)  
975.i2 975a5 \([1, 1, 0, -203125, -35321000]\) \(59319456301170001/594140625\) \(9283447265625\) \([2, 2]\) \(4608\) \(1.6476\)  
975.i8 975a6 \([1, 1, 0, 15125, -2468750]\) \(24487529386319/183539412225\) \(-2867803316015625\) \([2]\) \(4608\) \(1.6476\)  
975.i1 975a7 \([1, 1, 0, -3250000, -2256492875]\) \(242970740812818720001/24375\) \(380859375\) \([2]\) \(9216\) \(1.9941\)  
975.i3 975a8 \([1, 1, 0, -198250, -37090625]\) \(-55150149867714721/5950927734375\) \(-92983245849609375\) \([2]\) \(9216\) \(1.9941\)  

Rank

sage: E.rank()
 

The elliptic curves in class 975a have rank \(1\).

Complex multiplication

The elliptic curves in class 975a do not have complex multiplication.

Modular form 975.2.a.a

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

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

Isogeny graph

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

The vertices are labelled with Cremona labels.