Properties

 Label 2178c Number of curves 4 Conductor 2178 CM no Rank 1 Graph

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Show commands for: SageMath
sage: E = EllipticCurve("2178.b1")

sage: E.isogeny_class()

Elliptic curves in class 2178c

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
2178.b3 2178c1 [1, -1, 0, -6012, 170748] [2] 3840 $$\Gamma_0(N)$$-optimal
2178.b4 2178c2 [1, -1, 0, 4878, 713070] [2] 7680
2178.b1 2178c3 [1, -1, 0, -87687, -9934083] [2] 11520
2178.b2 2178c4 [1, -1, 0, -44127, -19839627] [2] 23040

Rank

sage: E.rank()

The elliptic curves in class 2178c have rank $$1$$.

Modular form2178.2.a.b

sage: E.q_eigenform(10)

$$q - q^{2} + q^{4} - 2q^{7} - q^{8} + 4q^{13} + 2q^{14} + q^{16} - 6q^{17} + 4q^{19} + O(q^{20})$$

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.