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SageMath
E = EllipticCurve("f1")
E.isogeny_class()
Elliptic curves in class 12075.f
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 12075.f1 | 12075d3 | \([1, 1, 1, -107338, 13490906]\) | \(8753151307882969/65205\) | \(1018828125\) | \([2]\) | \(33792\) | \(1.3231\) | |
| 12075.f2 | 12075d2 | \([1, 1, 1, -6713, 208406]\) | \(2141202151369/5832225\) | \(91128515625\) | \([2, 2]\) | \(16896\) | \(0.97648\) | |
| 12075.f3 | 12075d4 | \([1, 1, 1, -4088, 376406]\) | \(-483551781049/3672913125\) | \(-57389267578125\) | \([2]\) | \(33792\) | \(1.3231\) | |
| 12075.f4 | 12075d1 | \([1, 1, 1, -588, 156]\) | \(1439069689/828345\) | \(12942890625\) | \([2]\) | \(8448\) | \(0.62991\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 12075.f have rank \(1\).
Complex multiplication
The elliptic curves in class 12075.f do not have complex multiplication.Modular form 12075.2.a.f
sage: E.q_eigenform(10)
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.
