Properties

Label 317900e
Number of curves $4$
Conductor $317900$
CM no
Rank $1$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("e1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 317900e

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
317900.e4 317900e1 \([0, 1, 0, -327533, 70560688]\) \(643956736/15125\) \(91270182781250000\) \([2]\) \(4478976\) \(2.0399\) \(\Gamma_0(N)\)-optimal
317900.e3 317900e2 \([0, 1, 0, -724908, -134484812]\) \(436334416/171875\) \(16594578687500000000\) \([2]\) \(8957952\) \(2.3865\)  
317900.e2 317900e3 \([0, 1, 0, -3217533, -2194476812]\) \(610462990336/8857805\) \(53451469844011250000\) \([2]\) \(13436928\) \(2.5892\)  
317900.e1 317900e4 \([0, 1, 0, -51299908, -141441034812]\) \(154639330142416/33275\) \(3212710433900000000\) \([2]\) \(26873856\) \(2.9358\)  

Rank

sage: E.rank()
 

The elliptic curves in class 317900e have rank \(1\).

Complex multiplication

The elliptic curves in class 317900e do not have complex multiplication.

Modular form 317900.2.a.e

sage: E.q_eigenform(10)
 
\(q - 2q^{3} - 4q^{7} + q^{9} + q^{11} + 4q^{13} - 4q^{19} + O(q^{20})\)  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.

\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.