Show commands:
SageMath
E = EllipticCurve("r1")
E.isogeny_class()
Elliptic curves in class 26010r
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
26010.u6 | 26010r1 | \([1, -1, 0, -208134, -44852940]\) | \(-56667352321/16711680\) | \(-294063530918215680\) | \([2]\) | \(294912\) | \(2.0664\) | \(\Gamma_0(N)\)-optimal |
26010.u5 | 26010r2 | \([1, -1, 0, -3537414, -2559791052]\) | \(278202094583041/16646400\) | \(292914845250566400\) | \([2, 2]\) | \(589824\) | \(2.4129\) | |
26010.u4 | 26010r3 | \([1, -1, 0, -3745494, -2241553500]\) | \(330240275458561/67652010000\) | \(1190424238276130010000\) | \([2, 2]\) | \(1179648\) | \(2.7595\) | |
26010.u2 | 26010r4 | \([1, -1, 0, -56597814, -163874019132]\) | \(1139466686381936641/4080\) | \(71792854228080\) | \([2]\) | \(1179648\) | \(2.7595\) | |
26010.u7 | 26010r5 | \([1, -1, 0, 7959006, -13456805400]\) | \(3168685387909439/6278181696900\) | \(-110472692005622949516900\) | \([2]\) | \(2359296\) | \(3.1061\) | |
26010.u3 | 26010r6 | \([1, -1, 0, -18779274, 29338404768]\) | \(41623544884956481/2962701562500\) | \(52132549362222389062500\) | \([2, 2]\) | \(2359296\) | \(3.1061\) | |
26010.u8 | 26010r7 | \([1, -1, 0, 17036496, 127996524810]\) | \(31077313442863199/420227050781250\) | \(-7394436127312316894531250\) | \([2]\) | \(4718592\) | \(3.4527\) | |
26010.u1 | 26010r8 | \([1, -1, 0, -295135524, 1951617208518]\) | \(161572377633716256481/914742821250\) | \(16096077946613698571250\) | \([2]\) | \(4718592\) | \(3.4527\) |
Rank
sage: E.rank()
The elliptic curves in class 26010r have rank \(1\).
Complex multiplication
The elliptic curves in class 26010r do not have complex multiplication.Modular form 26010.2.a.r
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 4 & 8 & 8 & 16 & 16 \\ 2 & 1 & 2 & 2 & 4 & 4 & 8 & 8 \\ 4 & 2 & 1 & 4 & 2 & 2 & 4 & 4 \\ 4 & 2 & 4 & 1 & 8 & 8 & 16 & 16 \\ 8 & 4 & 2 & 8 & 1 & 4 & 8 & 8 \\ 8 & 4 & 2 & 8 & 4 & 1 & 2 & 2 \\ 16 & 8 & 4 & 16 & 8 & 2 & 1 & 4 \\ 16 & 8 & 4 & 16 & 8 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.