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SageMath
E = EllipticCurve("bp1")
E.isogeny_class()
Elliptic curves in class 94050bp
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
94050.cf7 | 94050bp1 | \([1, -1, 0, -19464192, -29001622784]\) | \(71595431380957421881/9522562500000000\) | \(108467938476562500000000\) | \([2]\) | \(12386304\) | \(3.1478\) | \(\Gamma_0(N)\)-optimal |
94050.cf5 | 94050bp2 | \([1, -1, 0, -300714192, -2007032872784]\) | \(264020672568758737421881/5803468580250000\) | \(66105134296910156250000\) | \([2, 2]\) | \(24772608\) | \(3.4944\) | |
94050.cf4 | 94050bp3 | \([1, -1, 0, -392823567, 2993013455341]\) | \(588530213343917460371881/861551575695360000\) | \(9813610916904960000000000\) | \([2]\) | \(37158912\) | \(3.6971\) | |
94050.cf6 | 94050bp4 | \([1, -1, 0, -290026692, -2156305185284]\) | \(-236859095231405581781881/39282983014374049500\) | \(-447457728398104407585937500\) | \([2]\) | \(49545216\) | \(3.8410\) | |
94050.cf2 | 94050bp5 | \([1, -1, 0, -4811401692, -128455135560284]\) | \(1081411559614045490773061881/522522049500\) | \(5951852720085937500\) | \([2]\) | \(49545216\) | \(3.8410\) | |
94050.cf3 | 94050bp6 | \([1, -1, 0, -508023567, 1094171855341]\) | \(1272998045160051207059881/691293848290254950400\) | \(7874268990681185294400000000\) | \([2, 2]\) | \(74317824\) | \(4.0437\) | |
94050.cf8 | 94050bp7 | \([1, -1, 0, 1961216433, 8608069175341]\) | \(73240740785321709623685719/45195275784938365817280\) | \(-514802438237813573137455000000\) | \([2]\) | \(148635648\) | \(4.3903\) | |
94050.cf1 | 94050bp8 | \([1, -1, 0, -4820463567, -127946970264659]\) | \(1087533321226184807035053481/8484255812957933638080\) | \(96640976369473962846255000000\) | \([2]\) | \(148635648\) | \(4.3903\) |
Rank
sage: E.rank()
The elliptic curves in class 94050bp have rank \(0\).
Complex multiplication
The elliptic curves in class 94050bp do not have complex multiplication.Modular form 94050.2.a.bp
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 3 & 4 & 4 & 6 & 12 & 12 \\ 2 & 1 & 6 & 2 & 2 & 3 & 6 & 6 \\ 3 & 6 & 1 & 12 & 12 & 2 & 4 & 4 \\ 4 & 2 & 12 & 1 & 4 & 6 & 3 & 12 \\ 4 & 2 & 12 & 4 & 1 & 6 & 12 & 3 \\ 6 & 3 & 2 & 6 & 6 & 1 & 2 & 2 \\ 12 & 6 & 4 & 3 & 12 & 2 & 1 & 4 \\ 12 & 6 & 4 & 12 & 3 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.