# Properties

 Label 38115m Number of curves $6$ Conductor $38115$ CM no Rank $1$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("m1")

sage: E.isogeny_class()

## Elliptic curves in class 38115m

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
38115.h6 38115m1 [1, -1, 1, 38092, -831265234] [2] 921600 $$\Gamma_0(N)$$-optimal
38115.h5 38115m2 [1, -1, 1, -13035353, -17790138088] [2, 2] 1843200
38115.h4 38115m3 [1, -1, 1, -27709628, 29613639872] [2] 3686400
38115.h2 38115m4 [1, -1, 1, -207536198, -1150718660044] [2, 2] 3686400
38115.h3 38115m5 [1, -1, 1, -206507093, -1162696207318] [2] 7372800
38115.h1 38115m6 [1, -1, 1, -3320578823, -73648500528094] [2] 7372800

## Rank

sage: E.rank()

The elliptic curves in class 38115m have rank $$1$$.

## Complex multiplication

The elliptic curves in class 38115m do not have complex multiplication.

## Modular form 38115.2.a.m

sage: E.q_eigenform(10)

$$q - q^{2} - q^{4} - q^{5} - q^{7} + 3q^{8} + q^{10} + 2q^{13} + q^{14} - q^{16} + 2q^{17} - 4q^{19} + O(q^{20})$$

## 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.

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

## Isogeny graph

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

The vertices are labelled with Cremona labels.