# Properties

 Label 53040b Number of curves $4$ Conductor $53040$ CM no Rank $2$ Graph # Related objects

Show commands: SageMath
sage: E = EllipticCurve("b1")

sage: E.isogeny_class()

## Elliptic curves in class 53040b

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
53040.j3 53040b1 $$[0, -1, 0, -476, -3264]$$ $$46689225424/7249905$$ $$1855975680$$ $$$$ $$24576$$ $$0.50172$$ $$\Gamma_0(N)$$-optimal
53040.j2 53040b2 $$[0, -1, 0, -2096, 34320]$$ $$994958062276/98903025$$ $$101276697600$$ $$[2, 2]$$ $$49152$$ $$0.84829$$
53040.j4 53040b3 $$[0, -1, 0, 2584, 161616]$$ $$931329171502/6107473125$$ $$-12508104960000$$ $$$$ $$98304$$ $$1.1949$$
53040.j1 53040b4 $$[0, -1, 0, -32696, 2286480]$$ $$1887517194957938/21849165$$ $$44747089920$$ $$$$ $$98304$$ $$1.1949$$

## Rank

sage: E.rank()

The elliptic curves in class 53040b have rank $$2$$.

## Complex multiplication

The elliptic curves in class 53040b do not have complex multiplication.

## Modular form 53040.2.a.b

sage: E.q_eigenform(10)

$$q - q^{3} - q^{5} + q^{9} - 4q^{11} - q^{13} + q^{15} + q^{17} - 8q^{19} + O(q^{20})$$ ## 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. 