Properties

Label 12075.v
Number of curves $4$
Conductor $12075$
CM no
Rank $0$
Graph

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Show commands: SageMath
E = EllipticCurve("v1")
 
E.isogeny_class()
 

Elliptic curves in class 12075.v

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
12075.v1 12075s3 \([1, 0, 1, -126276, 17260573]\) \(14251520160844849/264449745\) \(4132027265625\) \([2]\) \(46080\) \(1.5457\)  
12075.v2 12075s2 \([1, 0, 1, -8151, 250573]\) \(3832302404449/472410225\) \(7381409765625\) \([2, 2]\) \(23040\) \(1.1992\)  
12075.v3 12075s1 \([1, 0, 1, -2026, -31177]\) \(58818484369/7455105\) \(116486015625\) \([2]\) \(11520\) \(0.85260\) \(\Gamma_0(N)\)-optimal
12075.v4 12075s4 \([1, 0, 1, 11974, 1297073]\) \(12152722588271/53476250625\) \(-835566416015625\) \([2]\) \(46080\) \(1.5457\)  

Rank

sage: E.rank()
 

The elliptic curves in class 12075.v have rank \(0\).

Complex multiplication

The elliptic curves in class 12075.v do not have complex multiplication.

Modular form 12075.2.a.v

sage: E.q_eigenform(10)
 
\(q + q^{2} + q^{3} - q^{4} + q^{6} + q^{7} - 3 q^{8} + q^{9} + 4 q^{11} - q^{12} + 6 q^{13} + q^{14} - q^{16} + 2 q^{17} + q^{18} + O(q^{20})\) 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 LMFDB 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 LMFDB labels.