# Properties

 Label 13872bi Number of curves 6 Conductor 13872 CM no Rank 1 Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("13872.bm1")

sage: E.isogeny_class()

## Elliptic curves in class 13872bi

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
13872.bm5 13872bi1 [0, 1, 0, -157312, -22325068] [2] 110592 $$\Gamma_0(N)$$-optimal
13872.bm4 13872bi2 [0, 1, 0, -527232, 121351860] [2, 2] 221184
13872.bm2 13872bi3 [0, 1, 0, -8018112, 8735863860] [2, 2] 442368
13872.bm6 13872bi4 [0, 1, 0, 1044928, 708081972] [4] 442368
13872.bm1 13872bi5 [0, 1, 0, -128288352, 559236806388] [2] 884736
13872.bm3 13872bi6 [0, 1, 0, -7601952, 9683543412] [2] 884736

## Rank

sage: E.rank()

The elliptic curves in class 13872bi have rank $$1$$.

## Modular form 13872.2.a.bm

sage: E.q_eigenform(10)

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

## 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 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.