# Properties

 Label 115920.dp Number of curves $4$ Conductor $115920$ CM no Rank $1$ Graph

# Related objects

Show commands: SageMath
sage: E = EllipticCurve("dp1")

sage: E.isogeny_class()

## Elliptic curves in class 115920.dp

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
115920.dp1 115920bj4 $$[0, 0, 0, -875666307, -8747613671326]$$ $$49737293673675178002921218/6641736806881023047235$$ $$9916059918778912361337477120$$ $$[2]$$ $$70778880$$ $$4.0996$$
115920.dp2 115920bj2 $$[0, 0, 0, -845413707, -9461145444406]$$ $$89516703758060574923008036/1985322833430374025$$ $$1482035553864440488166400$$ $$[2, 2]$$ $$35389440$$ $$3.7531$$
115920.dp3 115920bj1 $$[0, 0, 0, -845409207, -9461251202506]$$ $$358061097267989271289240144/176126855625$$ $$32869498304160000$$ $$[2]$$ $$17694720$$ $$3.4065$$ $$\Gamma_0(N)$$-optimal
115920.dp4 115920bj3 $$[0, 0, 0, -815233107, -10167908699086]$$ $$-40133926989810174413190818/6689384645060302103835$$ $$-9987197759997870558608824320$$ $$[2]$$ $$70778880$$ $$4.0996$$

## Rank

sage: E.rank()

The elliptic curves in class 115920.dp have rank $$1$$.

## Complex multiplication

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

## Modular form 115920.2.a.dp

sage: E.q_eigenform(10)

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

## 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 LMFDB 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 LMFDB labels.