# Properties

 Label 4275.i Number of curves $3$ Conductor $4275$ CM no Rank $1$ Graph # Learn more about

Show commands for: SageMath
sage: E = EllipticCurve("i1")

sage: E.isogeny_class()

## Elliptic curves in class 4275.i

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
4275.i1 4275k3 $$[0, 0, 1, -173100, 27720031]$$ $$-50357871050752/19$$ $$-216421875$$ $$[]$$ $$7776$$ $$1.3875$$
4275.i2 4275k2 $$[0, 0, 1, -2100, 39406]$$ $$-89915392/6859$$ $$-78128296875$$ $$[]$$ $$2592$$ $$0.83816$$
4275.i3 4275k1 $$[0, 0, 1, 150, 31]$$ $$32768/19$$ $$-216421875$$ $$[]$$ $$864$$ $$0.28885$$ $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

The elliptic curves in class 4275.i have rank $$1$$.

## Complex multiplication

The elliptic curves in class 4275.i do not have complex multiplication.

## Modular form4275.2.a.i

sage: E.q_eigenform(10)

$$q - 2q^{4} + q^{7} - 3q^{11} + 4q^{13} + 4q^{16} - 3q^{17} + q^{19} + O(q^{20})$$ ## 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.

$$\left(\begin{array}{rrr} 1 & 3 & 9 \\ 3 & 1 & 3 \\ 9 & 3 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 