# Properties

 Label 7623i Number of curves 6 Conductor 7623 CM no Rank 1 Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("7623.f1")

sage: E.isogeny_class()

## Elliptic curves in class 7623i

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
7623.f4 7623i1 [1, -1, 1, -37049, 2753160]  19200 $$\Gamma_0(N)$$-optimal
7623.f3 7623i2 [1, -1, 1, -42494, 1895028] [2, 2] 38400
7623.f2 7623i3 [1, -1, 1, -309299, -64806222] [2, 2] 76800
7623.f6 7623i4 [1, -1, 1, 137191, 13610490]  76800
7623.f1 7623i5 [1, -1, 1, -4921214, -4200771594]  153600
7623.f5 7623i6 [1, -1, 1, 33736, -200922510]  153600

## Rank

sage: E.rank()

The elliptic curves in class 7623i have rank $$1$$.

## Modular form7623.2.a.f

sage: E.q_eigenform(10)

$$q - q^{2} - q^{4} + 2q^{5} - q^{7} + 3q^{8} - 2q^{10} - 6q^{13} + q^{14} - q^{16} + 2q^{17} - 4q^{19} + O(q^{20})$$

## 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.

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

## Isogeny graph

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

The vertices are labelled with Cremona labels. 