# Properties

 Label 115920ef Number of curves $4$ Conductor $115920$ CM no Rank $2$ Graph # Related objects

Show commands: SageMath
sage: E = EllipticCurve("ef1")

sage: E.isogeny_class()

## Elliptic curves in class 115920ef

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
115920.cn4 115920ef1 $$[0, 0, 0, 4173, -240014]$$ $$2691419471/9891840$$ $$-29536875970560$$ $$$$ $$294912$$ $$1.2676$$ $$\Gamma_0(N)$$-optimal
115920.cn3 115920ef2 $$[0, 0, 0, -41907, -2885006]$$ $$2725812332209/373262400$$ $$1114555554201600$$ $$[2, 2]$$ $$589824$$ $$1.6142$$
115920.cn2 115920ef3 $$[0, 0, 0, -174387, 25121266]$$ $$196416765680689/22365315000$$ $$66782472744960000$$ $$$$ $$1179648$$ $$1.9607$$
115920.cn1 115920ef4 $$[0, 0, 0, -646707, -200170766]$$ $$10017490085065009/235066440$$ $$701904628776960$$ $$$$ $$1179648$$ $$1.9607$$

## Rank

sage: E.rank()

The elliptic curves in class 115920ef have rank $$2$$.

## Complex multiplication

The elliptic curves in class 115920ef do not have complex multiplication.

## Modular form 115920.2.a.ef

sage: E.q_eigenform(10)

$$q + q^{5} - q^{7} - 4 q^{11} - 2 q^{13} - 6 q^{17} + 4 q^{19} + O(q^{20})$$

## 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.

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

## Isogeny graph

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

The vertices are labelled with Cremona labels. 