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SageMath
E = EllipticCurve("ee1")
E.isogeny_class()
Elliptic curves in class 248430ee
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 248430.ee6 | 248430ee1 | \([1, 0, 1, 124042, -6723832]\) | \(371694959/249600\) | \(-141740165309433600\) | \([2]\) | \(4128768\) | \(1.9798\) | \(\Gamma_0(N)\)-optimal |
| 248430.ee5 | 248430ee2 | \([1, 0, 1, -538438, -56012344]\) | \(30400540561/15210000\) | \(8637291323543610000\) | \([2, 2]\) | \(8257536\) | \(2.3264\) | |
| 248430.ee3 | 248430ee3 | \([1, 0, 1, -4678938, 3855932056]\) | \(19948814692561/231344100\) | \(131373201031098308100\) | \([2, 2]\) | \(16515072\) | \(2.6729\) | |
| 248430.ee2 | 248430ee4 | \([1, 0, 1, -6997618, -7119771592]\) | \(66730743078481/60937500\) | \(34604532546248437500\) | \([2]\) | \(16515072\) | \(2.6729\) | |
| 248430.ee1 | 248430ee5 | \([1, 0, 1, -74653388, 248262691016]\) | \(81025909800741361/11088090\) | \(6296585374863291690\) | \([2]\) | \(33030144\) | \(3.0195\) | |
| 248430.ee4 | 248430ee6 | \([1, 0, 1, -952488, 9830176696]\) | \(-168288035761/73415764890\) | \(-41690555496102208640490\) | \([2]\) | \(33030144\) | \(3.0195\) |
Rank
sage: E.rank()
The elliptic curves in class 248430ee have rank \(0\).
Complex multiplication
The elliptic curves in class 248430ee do not have complex multiplication.Modular form 248430.2.a.ee
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 Cremona numbering.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 2 & 2 \\ 4 & 2 & 4 & 1 & 8 & 8 \\ 8 & 4 & 2 & 8 & 1 & 4 \\ 8 & 4 & 2 & 8 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
