# Properties

 Label 304920ch Number of curves $6$ Conductor $304920$ CM no Rank $1$ Graph

# Learn more about

Show commands for: SageMath
sage: E = EllipticCurve("304920.ch1")

sage: E.isogeny_class()

## Elliptic curves in class 304920ch

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
304920.ch4 304920ch1 [0, 0, 0, -190938, 32113037] [2] 1310720 $$\Gamma_0(N)$$-optimal
304920.ch3 304920ch2 [0, 0, 0, -196383, 30184418] [2, 2] 2621440
304920.ch5 304920ch3 [0, 0, 0, 260997, 149926502] [2] 5242880
304920.ch2 304920ch4 [0, 0, 0, -740883, -212989282] [2, 2] 5242880
304920.ch6 304920ch5 [0, 0, 0, 1219317, -1148788762] [2] 10485760
304920.ch1 304920ch6 [0, 0, 0, -11413083, -14840306602] [2] 10485760

## Rank

sage: E.rank()

The elliptic curves in class 304920ch have rank $$1$$.

## Modular form 304920.2.a.ch

sage: E.q_eigenform(10)

$$q - q^{5} + q^{7} + 2q^{13} + 2q^{17} + 4q^{19} + O(q^{20})$$

## 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.