Show commands:
SageMath
E = EllipticCurve("d1")
E.isogeny_class()
Elliptic curves in class 41070.d
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
41070.d1 | 41070b4 | \([1, 1, 0, -466835873, 3882153327477]\) | \(4385367890843575421521/24975000000\) | \(64079017064775000000\) | \([2]\) | \(9455616\) | \(3.4123\) | |
41070.d2 | 41070b6 | \([1, 1, 0, -414978153, -3239790411267]\) | \(3080272010107543650001/15465841417699560\) | \(39681117762797761119680040\) | \([2]\) | \(18911232\) | \(3.7589\) | |
41070.d3 | 41070b3 | \([1, 1, 0, -40145953, 10979326453]\) | \(2788936974993502801/1593609593601600\) | \(4088766219939382544654400\) | \([2, 2]\) | \(9455616\) | \(3.4123\) | |
41070.d4 | 41070b2 | \([1, 1, 0, -29193953, 60576553653]\) | \(1072487167529950801/2554882560000\) | \(6555129656085527040000\) | \([2, 2]\) | \(4727808\) | \(3.0657\) | |
41070.d5 | 41070b1 | \([1, 1, 0, -1156833, 1648134837]\) | \(-66730743078481/419010969600\) | \(-1075067510363416166400\) | \([2]\) | \(2363904\) | \(2.7191\) | \(\Gamma_0(N)\)-optimal |
41070.d6 | 41070b5 | \([1, 1, 0, 159454247, 87745563373]\) | \(174751791402194852399/102423900876336360\) | \(-262791707391214442018931240\) | \([2]\) | \(18911232\) | \(3.7589\) |
Rank
sage: E.rank()
The elliptic curves in class 41070.d have rank \(1\).
Complex multiplication
The elliptic curves in class 41070.d do not have complex multiplication.Modular form 41070.2.a.d
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.
\(\left(\begin{array}{rrrrrr} 1 & 8 & 4 & 2 & 4 & 8 \\ 8 & 1 & 2 & 4 & 8 & 4 \\ 4 & 2 & 1 & 2 & 4 & 2 \\ 2 & 4 & 2 & 1 & 2 & 4 \\ 4 & 8 & 4 & 2 & 1 & 8 \\ 8 & 4 & 2 & 4 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.