Properties

 Label 388416.ba Number of curves $6$ Conductor $388416$ CM no Rank $1$ Graph

Related objects

Show commands for: SageMath
sage: E = EllipticCurve("388416.ba1")

sage: E.isogeny_class()

Elliptic curves in class 388416.ba

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
388416.ba1 388416ba6 [0, -1, 0, -253739299969, -49195904458658687] [2] 1698693120
388416.ba2 388416ba4 [0, -1, 0, -15888878209, -765610271321471] [2, 2] 849346560
388416.ba3 388416ba5 [0, -1, 0, -5412003969, -1760186229049215] [2] 1698693120
388416.ba4 388416ba2 [0, -1, 0, -1678031489, 6649771983489] [2, 2] 424673280
388416.ba5 388416ba1 [0, -1, 0, -1299233409, 18003032277633] [2] 212336640 $$\Gamma_0(N)$$-optimal
388416.ba6 388416ba3 [0, -1, 0, 6472045951, 52295095693953] [2] 849346560

Rank

sage: E.rank()

The elliptic curves in class 388416.ba have rank $$1$$.

Modular form 388416.2.a.ba

sage: E.q_eigenform(10)

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

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 LMFDB numbering.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.