Properties

Label 12100f
Number of curves $4$
Conductor $12100$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("f1")
 
E.isogeny_class()
 

Elliptic curves in class 12100f

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
12100.j3 12100f1 \([0, -1, 0, -4033, -66438]\) \(16384/5\) \(2214451250000\) \([2]\) \(17280\) \(1.0737\) \(\Gamma_0(N)\)-optimal
12100.j4 12100f2 \([0, -1, 0, 11092, -459688]\) \(21296/25\) \(-177156100000000\) \([2]\) \(34560\) \(1.4203\)  
12100.j1 12100f3 \([0, -1, 0, -125033, 17055062]\) \(488095744/125\) \(55361281250000\) \([2]\) \(51840\) \(1.6230\)  
12100.j2 12100f4 \([0, -1, 0, -109908, 21320312]\) \(-20720464/15625\) \(-110722562500000000\) \([2]\) \(103680\) \(1.9696\)  

Rank

sage: E.rank()
 

The elliptic curves in class 12100f have rank \(1\).

Complex multiplication

The elliptic curves in class 12100f do not have complex multiplication.

Modular form 12100.2.a.f

sage: E.q_eigenform(10)
 
\(q + 2 q^{3} + 2 q^{7} + q^{9} + 2 q^{13} - 6 q^{17} + 4 q^{19} + O(q^{20})\) 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.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.