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Properties

Label 33135.d
Number of curves $8$
Conductor $33135$
CM no
Rank $0$
Graph

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Show commands: SageMath
E = EllipticCurve("d1")
 
E.isogeny_class()
 

Elliptic curves in class 33135.d

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
33135.d1 33135a8 \([1, 1, 1, -4771486, 4009713254]\) \(1114544804970241/405\) \(4365582208245\) \([2]\) \(412160\) \(2.2159\)  
33135.d2 33135a6 \([1, 1, 1, -298261, 62539514]\) \(272223782641/164025\) \(1768060794339225\) \([2, 2]\) \(206080\) \(1.8694\)  
33135.d3 33135a7 \([1, 1, 1, -243036, 86485074]\) \(-147281603041/215233605\) \(-2320049374331931045\) \([2]\) \(412160\) \(2.2159\)  
33135.d4 33135a4 \([1, 1, 1, -176766, -28678932]\) \(56667352321/15\) \(161688229935\) \([2]\) \(103040\) \(1.5228\)  
33135.d5 33135a3 \([1, 1, 1, -22136, 577064]\) \(111284641/50625\) \(545697776030625\) \([2, 2]\) \(103040\) \(1.5228\)  
33135.d6 33135a2 \([1, 1, 1, -11091, -447912]\) \(13997521/225\) \(2425323449025\) \([2, 2]\) \(51520\) \(1.1762\)  
33135.d7 33135a1 \([1, 1, 1, -46, -19366]\) \(-1/15\) \(-161688229935\) \([2]\) \(25760\) \(0.82965\) \(\Gamma_0(N)\)-optimal
33135.d8 33135a5 \([1, 1, 1, 77269, 4433978]\) \(4733169839/3515625\) \(-37895678891015625\) \([2]\) \(206080\) \(1.8694\)  

Rank

sage: E.rank()
 

The elliptic curves in class 33135.d have rank \(0\).

Complex multiplication

The elliptic curves in class 33135.d do not have complex multiplication.

Modular form 33135.2.a.d

sage: E.q_eigenform(10)
 
\(q - q^{2} - q^{3} - q^{4} - q^{5} + q^{6} + 3 q^{8} + q^{9} + q^{10} + 4 q^{11} + q^{12} + 2 q^{13} + q^{15} - q^{16} + 2 q^{17} - q^{18} - 4 q^{19} + O(q^{20})\) 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 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.