Properties

Label 270400ii
Number of curves $2$
Conductor $270400$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("ii1")
 
E.isogeny_class()
 

Elliptic curves in class 270400ii

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
270400.ii1 270400ii1 \([0, -1, 0, -227412033, -1319726704063]\) \(65787589563409/10400000\) \(205614340505600000000000\) \([2]\) \(61931520\) \(3.4839\) \(\Gamma_0(N)\)-optimal
270400.ii2 270400ii2 \([0, -1, 0, -205780033, -1580889840063]\) \(-48743122863889/26406250000\) \(-522067661440000000000000000\) \([2]\) \(123863040\) \(3.8304\)  

Rank

sage: E.rank()
 

The elliptic curves in class 270400ii have rank \(1\).

Complex multiplication

The elliptic curves in class 270400ii do not have complex multiplication.

Modular form 270400.2.a.ii

sage: E.q_eigenform(10)
 
\(q + 2 q^{3} - 4 q^{7} + q^{9} - 2 q^{11} - 2 q^{17} + 6 q^{19} + O(q^{20})\) 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.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.