Properties

Label 11025z
Number of curves $8$
Conductor $11025$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("z1")
 
E.isogeny_class()
 

Elliptic curves in class 11025z

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
11025.p7 11025z1 \([1, -1, 1, -230, 215772]\) \(-1/15\) \(-20101434609375\) \([2]\) \(18432\) \(1.2316\) \(\Gamma_0(N)\)-optimal
11025.p6 11025z2 \([1, -1, 1, -55355, 4956522]\) \(13997521/225\) \(301521519140625\) \([2, 2]\) \(36864\) \(1.5781\)  
11025.p5 11025z3 \([1, -1, 1, -110480, -6509478]\) \(111284641/50625\) \(67842341806640625\) \([2, 2]\) \(73728\) \(1.9247\)  
11025.p4 11025z4 \([1, -1, 1, -882230, 319169022]\) \(56667352321/15\) \(20101434609375\) \([2]\) \(73728\) \(1.9247\)  
11025.p2 11025z5 \([1, -1, 1, -1488605, -698328228]\) \(272223782641/164025\) \(219809187453515625\) \([2, 2]\) \(147456\) \(2.2713\)  
11025.p8 11025z6 \([1, -1, 1, 385645, -49176228]\) \(4733169839/3515625\) \(-4711273736572265625\) \([2]\) \(147456\) \(2.2713\)  
11025.p1 11025z7 \([1, -1, 1, -23814230, -44724460728]\) \(1114544804970241/405\) \(542738734453125\) \([2]\) \(294912\) \(2.6179\)  
11025.p3 11025z8 \([1, -1, 1, -1212980, -965133228]\) \(-147281603041/215233605\) \(-288433615776503203125\) \([2]\) \(294912\) \(2.6179\)  

Rank

sage: E.rank()
 

The elliptic curves in class 11025z have rank \(1\).

Complex multiplication

The elliptic curves in class 11025z do not have complex multiplication.

Modular form 11025.2.a.z

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