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Properties

Label 12075b
Number of curves $2$
Conductor $12075$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("b1")
 
E.isogeny_class()
 

Elliptic curves in class 12075b

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
12075.q1 12075b1 \([1, 1, 0, -5750, -146625]\) \(1345938541921/203765625\) \(3183837890625\) \([2]\) \(18432\) \(1.1231\) \(\Gamma_0(N)\)-optimal
12075.q2 12075b2 \([1, 1, 0, 9875, -787250]\) \(6814692748079/21258460125\) \(-332163439453125\) \([2]\) \(36864\) \(1.4696\)  

Rank

sage: E.rank()
 

The elliptic curves in class 12075b have rank \(1\).

Complex multiplication

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

Modular form 12075.2.a.b

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