Properties

 Label 1680m Number of curves 8 Conductor 1680 CM no Rank 1 Graph

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Show commands for: SageMath
sage: E = EllipticCurve("1680.g1")

sage: E.isogeny_class()

Elliptic curves in class 1680m

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
1680.g7 1680m1 [0, -1, 0, -7960, -270608] [2] 2304 $$\Gamma_0(N)$$-optimal
1680.g6 1680m2 [0, -1, 0, -9240, -176400] [2, 2] 4608
1680.g5 1680m3 [0, -1, 0, -23560, 1065712] [2] 6912
1680.g4 1680m4 [0, -1, 0, -69720, 6984432] [4] 9216
1680.g8 1680m5 [0, -1, 0, 30760, -1328400] [2] 9216
1680.g2 1680m6 [0, -1, 0, -351240, 80233200] [2, 2] 13824
1680.g1 1680m7 [0, -1, 0, -5619720, 5129544432] [4] 27648
1680.g3 1680m8 [0, -1, 0, -325640, 92398320] [2] 27648

Rank

sage: E.rank()

The elliptic curves in class 1680m have rank $$1$$.

Modular form1680.2.a.g

sage: E.q_eigenform(10)

$$q - q^{3} + q^{5} - q^{7} + q^{9} + 2q^{13} - q^{15} - 6q^{17} - 8q^{19} + O(q^{20})$$

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.

$$\left(\begin{array}{rrrrrrrr} 1 & 2 & 3 & 4 & 4 & 6 & 12 & 12 \\ 2 & 1 & 6 & 2 & 2 & 3 & 6 & 6 \\ 3 & 6 & 1 & 12 & 12 & 2 & 4 & 4 \\ 4 & 2 & 12 & 1 & 4 & 6 & 3 & 12 \\ 4 & 2 & 12 & 4 & 1 & 6 & 12 & 3 \\ 6 & 3 & 2 & 6 & 6 & 1 & 2 & 2 \\ 12 & 6 & 4 & 3 & 12 & 2 & 1 & 4 \\ 12 & 6 & 4 & 12 & 3 & 2 & 4 & 1 \end{array}\right)$$

Isogeny graph

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

The vertices are labelled with Cremona labels.