Properties

 Label 33135a Number of curves 8 Conductor 33135 CM no Rank 0 Graph

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Show commands for: SageMath
sage: E = EllipticCurve("33135.d1")

sage: E.isogeny_class()

Elliptic curves in class 33135a

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
33135.d7 33135a1 [1, 1, 1, -46, -19366] [2] 25760 $$\Gamma_0(N)$$-optimal
33135.d6 33135a2 [1, 1, 1, -11091, -447912] [2, 2] 51520
33135.d5 33135a3 [1, 1, 1, -22136, 577064] [2, 2] 103040
33135.d4 33135a4 [1, 1, 1, -176766, -28678932] [2] 103040
33135.d8 33135a5 [1, 1, 1, 77269, 4433978] [2] 206080
33135.d2 33135a6 [1, 1, 1, -298261, 62539514] [2, 2] 206080
33135.d3 33135a7 [1, 1, 1, -243036, 86485074] [2] 412160
33135.d1 33135a8 [1, 1, 1, -4771486, 4009713254] [2] 412160

Rank

sage: E.rank()

The elliptic curves in class 33135a have rank $$0$$.

Modular form 33135.2.a.d

sage: E.q_eigenform(10)

$$q - q^{2} - q^{3} - q^{4} - q^{5} + q^{6} + 3q^{8} + q^{9} + q^{10} + 4q^{11} + q^{12} + 2q^{13} + q^{15} - q^{16} + 2q^{17} - q^{18} - 4q^{19} + O(q^{20})$$

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.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.