# Properties

 Label 7728.t Number of curves $2$ Conductor $7728$ CM no Rank $1$ Graph # Related objects

Show commands: SageMath
sage: E = EllipticCurve("t1")

sage: E.isogeny_class()

## Elliptic curves in class 7728.t

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
7728.t1 7728p2 $$[0, 1, 0, -4004232, -3085271820]$$ $$1733490909744055732873/99355964553216$$ $$406962030809972736$$ $$$$ $$202752$$ $$2.4426$$
7728.t2 7728p1 $$[0, 1, 0, -235912, -54035212]$$ $$-354499561600764553/101902222098432$$ $$-417391501715177472$$ $$$$ $$101376$$ $$2.0961$$ $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

The elliptic curves in class 7728.t have rank $$1$$.

## Complex multiplication

The elliptic curves in class 7728.t do not have complex multiplication.

## Modular form7728.2.a.t

sage: E.q_eigenform(10)

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

## 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.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 