# Properties

 Label 3025b Number of curves $4$ Conductor $3025$ CM no Rank $0$ Graph # Learn more about

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

sage: E.isogeny_class()

## Elliptic curves in class 3025b

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
3025.f4 3025b1 $$[1, -1, 0, 2458, -37009]$$ $$59319/55$$ $$-1522435234375$$ $$$$ $$2880$$ $$1.0249$$ $$\Gamma_0(N)$$-optimal
3025.f3 3025b2 $$[1, -1, 0, -12667, -324384]$$ $$8120601/3025$$ $$83733937890625$$ $$[2, 2]$$ $$5760$$ $$1.3714$$
3025.f1 3025b3 $$[1, -1, 0, -179042, -29107259]$$ $$22930509321/6875$$ $$190304404296875$$ $$$$ $$11520$$ $$1.7180$$
3025.f2 3025b4 $$[1, -1, 0, -88292, 9884991]$$ $$2749884201/73205$$ $$2026361296953125$$ $$$$ $$11520$$ $$1.7180$$

## Rank

sage: E.rank()

The elliptic curves in class 3025b have rank $$0$$.

## Complex multiplication

The elliptic curves in class 3025b do not have complex multiplication.

## Modular form3025.2.a.b

sage: E.q_eigenform(10)

$$q + q^{2} - q^{4} - 3q^{8} - 3q^{9} + 2q^{13} - q^{16} + 6q^{17} - 3q^{18} + 4q^{19} + O(q^{20})$$ ## 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.

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

## Isogeny graph

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

The vertices are labelled with Cremona labels. 