# Properties

 Label 5712t Number of curves $6$ Conductor $5712$ CM no Rank $0$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("5712.o1")

sage: E.isogeny_class()

## Elliptic curves in class 5712t

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
5712.o5 5712t1 [0, 1, 0, -1123904, -458409420] [2] 92160 $$\Gamma_0(N)$$-optimal
5712.o4 5712t2 [0, 1, 0, -1451584, -169657804] [2, 2] 184320
5712.o2 5712t3 [0, 1, 0, -13744704, 19474747956] [2, 4] 368640
5712.o6 5712t4 [0, 1, 0, 5598656, -1328717260] [2] 368640
5712.o1 5712t5 [0, 1, 0, -219497664, 1251605773620] [8] 737280
5712.o3 5712t6 [0, 1, 0, -4681664, 44782380852] [4] 737280

## Rank

sage: E.rank()

The elliptic curves in class 5712t have rank $$0$$.

## Modular form5712.2.a.o

sage: E.q_eigenform(10)

$$q + q^{3} - 2q^{5} - q^{7} + q^{9} - 4q^{11} - 2q^{13} - 2q^{15} + q^{17} - 4q^{19} + O(q^{20})$$

## 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.