Properties

 Label 6800t Number of curves 4 Conductor 6800 CM no Rank 1 Graph

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Show commands for: SageMath
sage: E = EllipticCurve("6800.b1")

sage: E.isogeny_class()

Elliptic curves in class 6800t

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
6800.b4 6800t1 [0, 1, 0, -1208, -10412] [2] 6912 $$\Gamma_0(N)$$-optimal
6800.b3 6800t2 [0, 1, 0, -17208, -874412] [2] 13824
6800.b2 6800t3 [0, 1, 0, -41208, 3205588] [2] 20736
6800.b1 6800t4 [0, 1, 0, -45208, 2541588] [2] 41472

Rank

sage: E.rank()

The elliptic curves in class 6800t have rank $$1$$.

Modular form6800.2.a.b

sage: E.q_eigenform(10)

$$q - 2q^{3} - 4q^{7} + q^{9} - 6q^{11} - 2q^{13} + q^{17} + 4q^{19} + O(q^{20})$$

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.

$$\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)$$

Isogeny graph

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

The vertices are labelled with Cremona labels.