# Properties

 Label 97461.s Number of curves $6$ Conductor $97461$ CM no Rank $0$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("s1")

sage: E.isogeny_class()

## Elliptic curves in class 97461.s

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
97461.s1 97461l6 $$[1, -1, 0, -17895348, -29112590781]$$ $$7389727131216686257/6115533215337$$ $$524505561726112197777$$ $$[2]$$ $$4718592$$ $$2.9041$$
97461.s2 97461l4 $$[1, -1, 0, -1364463, -239747040]$$ $$3275619238041697/1605271262049$$ $$137677889298717241929$$ $$[2, 2]$$ $$2359296$$ $$2.5576$$
97461.s3 97461l2 $$[1, -1, 0, -727218, 236274975]$$ $$495909170514577/6224736609$$ $$533871513200623689$$ $$[2, 2]$$ $$1179648$$ $$2.2110$$
97461.s4 97461l1 $$[1, -1, 0, -725013, 237792456]$$ $$491411892194497/78897$$ $$6766689648537$$ $$[2]$$ $$589824$$ $$1.8644$$ $$\Gamma_0(N)$$-optimal
97461.s5 97461l3 $$[1, -1, 0, -125253, 615151746]$$ $$-2533811507137/1904381781393$$ $$-163331438293147586553$$ $$[2]$$ $$2359296$$ $$2.5576$$
97461.s6 97461l5 $$[1, -1, 0, 4970502, -1839959199]$$ $$158346567380527343/108665074944153$$ $$-9319781966134294440513$$ $$[2]$$ $$4718592$$ $$2.9041$$

## Rank

sage: E.rank()

The elliptic curves in class 97461.s have rank $$0$$.

## Complex multiplication

The elliptic curves in class 97461.s do not have complex multiplication.

## Modular form 97461.2.a.s

sage: E.q_eigenform(10)

$$q + q^{2} - q^{4} - 2q^{5} - 3q^{8} - 2q^{10} + 4q^{11} - q^{13} - q^{16} + q^{17} + 4q^{19} + O(q^{20})$$

## 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 LMFDB numbering.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels.