# Properties

 Label 75504bj Number of curves $6$ Conductor $75504$ CM no Rank $1$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("75504.j1")

sage: E.isogeny_class()

## Elliptic curves in class 75504bj

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
75504.j5 75504bj1 [0, -1, 0, -46504, 5552560] [2] 491520 $$\Gamma_0(N)$$-optimal
75504.j4 75504bj2 [0, -1, 0, -830584, 291584944] [2, 2] 983040
75504.j3 75504bj3 [0, -1, 0, -917704, 226767664] [2, 2] 1966080
75504.j1 75504bj4 [0, -1, 0, -13288744, 18649929520] [2] 1966080
75504.j6 75504bj5 [0, -1, 0, 2596136, 1542349360] [2] 3932160
75504.j2 75504bj6 [0, -1, 0, -5825464, -5238513872] [2] 3932160

## Rank

sage: E.rank()

The elliptic curves in class 75504bj have rank $$1$$.

## Modular form 75504.2.a.j

sage: E.q_eigenform(10)

$$q - q^{3} - 2q^{5} + q^{9} - q^{13} + 2q^{15} + 6q^{17} - 4q^{19} + O(q^{20})$$

## 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.

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

## Isogeny graph

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

The vertices are labelled with Cremona labels.