Properties

 Label 38976x Number of curves $6$ Conductor $38976$ CM no Rank $1$ Graph

Related objects

Show commands for: SageMath
sage: E = EllipticCurve("38976.bw1")

sage: E.isogeny_class()

Elliptic curves in class 38976x

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
38976.bw5 38976x1 [0, 1, 0, -50177, 4615263] [2] 196608 $$\Gamma_0(N)$$-optimal
38976.bw4 38976x2 [0, 1, 0, -818497, 284744735] [2, 2] 393216
38976.bw3 38976x3 [0, 1, 0, -834177, 273251295] [2, 2] 786432
38976.bw1 38976x4 [0, 1, 0, -13095937, 18236817503] [2] 786432
38976.bw6 38976x5 [0, 1, 0, 798783, 1214162847] [2] 1572864
38976.bw2 38976x6 [0, 1, 0, -2718017, -1402989537] [2] 1572864

Rank

sage: E.rank()

The elliptic curves in class 38976x have rank $$1$$.

Modular form 38976.2.a.bw

sage: E.q_eigenform(10)

$$q + q^{3} + 2q^{5} + q^{7} + q^{9} - 4q^{11} + 2q^{13} + 2q^{15} + 2q^{17} + 4q^{19} + O(q^{20})$$

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.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.