Properties

 Label 5850bm Number of curves $6$ Conductor $5850$ CM no Rank $1$ Graph

Related objects

Show commands for: SageMath
sage: E = EllipticCurve("5850.bo1")

sage: E.isogeny_class()

Elliptic curves in class 5850bm

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
5850.bo6 5850bm1 [1, -1, 1, 3370, -31003] [2] 12288 $$\Gamma_0(N)$$-optimal
5850.bo5 5850bm2 [1, -1, 1, -14630, -247003] [2, 2] 24576
5850.bo2 5850bm3 [1, -1, 1, -190130, -31837003] [2] 49152
5850.bo3 5850bm4 [1, -1, 1, -127130, 17302997] [2, 2] 49152
5850.bo1 5850bm5 [1, -1, 1, -2028380, 1112422997] [2] 98304
5850.bo4 5850bm6 [1, -1, 1, -25880, 44032997] [2] 98304

Rank

sage: E.rank()

The elliptic curves in class 5850bm have rank $$1$$.

Modular form5850.2.a.bo

sage: E.q_eigenform(10)

$$q + q^{2} + q^{4} + q^{8} - 4q^{11} - q^{13} + q^{16} - 6q^{17} + 4q^{19} + O(q^{20})$$

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.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.