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SageMath
E = EllipticCurve("gb1")
E.isogeny_class()
Elliptic curves in class 76050.gb
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 76050.gb1 | 76050ew4 | \([1, -1, 1, -3290247005, -72641691675003]\) | \(71647584155243142409/10140000\) | \(557500964633437500000\) | \([2]\) | \(41287680\) | \(3.8338\) | |
| 76050.gb2 | 76050ew3 | \([1, -1, 1, -236079005, -776972619003]\) | \(26465989780414729/10571870144160\) | \(581245345497893290402500000\) | \([2]\) | \(41287680\) | \(3.8338\) | |
| 76050.gb3 | 76050ew2 | \([1, -1, 1, -205659005, -1134772659003]\) | \(17496824387403529/6580454400\) | \(361795826008515600000000\) | \([2, 2]\) | \(20643840\) | \(3.4872\) | |
| 76050.gb4 | 76050ew1 | \([1, -1, 1, -10971005, -23104179003]\) | \(-2656166199049/2658140160\) | \(-146145532872867840000000\) | \([2]\) | \(10321920\) | \(3.1406\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 76050.gb have rank \(1\).
Complex multiplication
The elliptic curves in class 76050.gb do not have complex multiplication.Modular form 76050.2.a.gb
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.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
