Show commands:
SageMath
E = EllipticCurve("r1")
E.isogeny_class()
Elliptic curves in class 100800.r
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
100800.r1 | 100800mi4 | \([0, 0, 0, -91854000300, -10715075262502000]\) | \(229625675762164624948320008/9568125\) | \(3571283520000000000\) | \([2]\) | \(165150720\) | \(4.4606\) | |
100800.r2 | 100800mi2 | \([0, 0, 0, -5740875300, -167423033752000]\) | \(448487713888272974160064/91549016015625\) | \(4271310891225000000000000\) | \([2, 2]\) | \(82575360\) | \(4.1140\) | |
100800.r3 | 100800mi3 | \([0, 0, 0, -5721192300, -168628066378000]\) | \(-55486311952875723077768/801237030029296875\) | \(-299060118984375000000000000000\) | \([2]\) | \(165150720\) | \(4.4606\) | |
100800.r4 | 100800mi1 | \([0, 0, 0, -360035175, -2597139043000]\) | \(7079962908642659949376/100085966990454375\) | \(72962669936041239375000000\) | \([2]\) | \(41287680\) | \(3.7674\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 100800.r have rank \(1\).
Complex multiplication
The elliptic curves in class 100800.r do not have complex multiplication.Modular form 100800.2.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 LMFDB numbering.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.