Properties

Label 257400ej
Number of curves $4$
Conductor $257400$
CM no
Rank $1$
Graph

Related objects

Downloads

Learn more

Show commands: SageMath
sage: E = EllipticCurve("ej1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 257400ej

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
257400.ej4 257400ej1 \([0, 0, 0, -20555175, -48828311750]\) \(-329381898333928144/162600887109375\) \(-474144186810937500000000\) \([2]\) \(33030144\) \(3.2482\) \(\Gamma_0(N)\)-optimal
257400.ej3 257400ej2 \([0, 0, 0, -360867675, -2638266124250]\) \(445574312599094932036/61129333175625\) \(713012542160490000000000\) \([2, 2]\) \(66060288\) \(3.5948\)  
257400.ej2 257400ej3 \([0, 0, 0, -393042675, -2139843199250]\) \(287849398425814280018/81784533026485575\) \(1907869586441855493600000000\) \([2]\) \(132120576\) \(3.9414\)  
257400.ej1 257400ej4 \([0, 0, 0, -5773692675, -168860709049250]\) \(912446049969377120252018/17177299425\) \(400712040986400000000\) \([2]\) \(132120576\) \(3.9414\)  

Rank

sage: E.rank()
 

The elliptic curves in class 257400ej have rank \(1\).

Complex multiplication

The elliptic curves in class 257400ej do not have complex multiplication.

Modular form 257400.2.a.ej

sage: E.q_eigenform(10)
 
\(q + 4 q^{7} + q^{11} - q^{13} - 2 q^{17} + 4 q^{19} + O(q^{20})\) Copy content Toggle raw display

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.