# Properties

 Label 97104cb Number of curves $6$ Conductor $97104$ CM no Rank $1$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("97104.bf1")

sage: E.isogeny_class()

## Elliptic curves in class 97104cb

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
97104.bf5 97104cb1 [0, -1, 0, -324808352, -2250216630528] [2] 26542080 $$\Gamma_0(N)$$-optimal
97104.bf4 97104cb2 [0, -1, 0, -419507872, -831011744000] [2, 2] 53084160
97104.bf6 97104cb3 [0, -1, 0, 1618011488, -6537695967488] [4] 106168320
97104.bf2 97104cb4 [0, -1, 0, -3972219552, 95703270024960] [2, 2] 106168320
97104.bf3 97104cb5 [0, -1, 0, -1353000992, 220023955131648] [2] 212336640
97104.bf1 97104cb6 [0, -1, 0, -63434824992, 6149519774744832] [2] 212336640

## Rank

sage: E.rank()

The elliptic curves in class 97104cb have rank $$1$$.

## Modular form 97104.2.a.bf

sage: E.q_eigenform(10)

$$q - q^{3} + 2q^{5} + q^{7} + q^{9} + 4q^{11} - 2q^{13} - 2q^{15} - 4q^{19} + O(q^{20})$$

## 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.

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

## Isogeny graph

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

The vertices are labelled with Cremona labels.