# Properties

 Label 245.c Number of curves $3$ Conductor $245$ CM no Rank $1$ Graph # Related objects

Show commands: SageMath
sage: E = EllipticCurve("c1")

sage: E.isogeny_class()

## Elliptic curves in class 245.c

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
245.c1 245c3 $$[0, -1, 1, -6435, 210006]$$ $$-250523582464/13671875$$ $$-1608482421875$$ $$[]$$ $$288$$ $$1.1004$$
245.c2 245c1 $$[0, -1, 1, -65, -204]$$ $$-262144/35$$ $$-4117715$$ $$[]$$ $$32$$ $$0.0018049$$ $$\Gamma_0(N)$$-optimal
245.c3 245c2 $$[0, -1, 1, 425, 433]$$ $$71991296/42875$$ $$-5044200875$$ $$[]$$ $$96$$ $$0.55111$$

## Rank

sage: E.rank()

The elliptic curves in class 245.c have rank $$1$$.

## Complex multiplication

The elliptic curves in class 245.c do not have complex multiplication.

## Modular form245.2.a.c

sage: E.q_eigenform(10)

$$q - q^{3} - 2q^{4} + q^{5} - 2q^{9} - 3q^{11} + 2q^{12} - 5q^{13} - q^{15} + 4q^{16} - 3q^{17} - 2q^{19} + O(q^{20})$$ ## 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.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 