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SageMath
E = EllipticCurve("r1")
E.isogeny_class()
Elliptic curves in class 188370.r
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 188370.r1 | 188370cc8 | \([1, -1, 0, -387520186950, -92837587876005600]\) | \(8828342566147309471108534663879471201/1536341563898865415843582949700\) | \(1119993000082272888149971970331300\) | \([6]\) | \(1826095104\) | \(5.3471\) | |
| 188370.r2 | 188370cc5 | \([1, -1, 0, -387504000450, -92845732302067500]\) | \(8827236347661221188886967161105287201/46539238473000000\) | \(33927104846817000000\) | \([2]\) | \(608698368\) | \(4.7978\) | |
| 188370.r3 | 188370cc7 | \([1, -1, 0, -166229471070, 25212768508864296]\) | \(696819431300451649932999125896765921/26551778102890598266349123437500\) | \(19356246237007246136168510985937500\) | \([6]\) | \(1826095104\) | \(5.3471\) | |
| 188370.r4 | 188370cc6 | \([1, -1, 0, -26653050450, -1141465011637500]\) | \(2872347286043717137884962530087201/890114999660918118510786090000\) | \(648893834752809308394363059610000\) | \([2, 6]\) | \(913047552\) | \(5.0006\) | |
| 188370.r5 | 188370cc4 | \([1, -1, 0, -24268533570, -1444477923948204]\) | \(2168337038351679228688694521765921/18360082088470458984375000000\) | \(13384499842494964599609375000000\) | \([2]\) | \(608698368\) | \(4.7978\) | |
| 188370.r6 | 188370cc2 | \([1, -1, 0, -24219000450, -1450709973067500]\) | \(2155087111607167363355460545287201/156481162929000000000000\) | \(114074767775241000000000000\) | \([2, 2]\) | \(304349184\) | \(4.4512\) | |
| 188370.r7 | 188370cc1 | \([1, -1, 0, -1510592130, -22764382770924]\) | \(-522923112164227281987660878881/4484275679769919488000000\) | \(-3269036970552271306752000000\) | \([2]\) | \(152174592\) | \(4.1047\) | \(\Gamma_0(N)\)-optimal |
| 188370.r8 | 188370cc3 | \([1, -1, 0, 4624671870, -120503855212524]\) | \(15005102139088880168192111025119/17288486242801155608083603200\) | \(-12603306471002042438292946732800\) | \([6]\) | \(456523776\) | \(4.6540\) |
Rank
sage: E.rank()
The elliptic curves in class 188370.r have rank \(1\).
Complex multiplication
The elliptic curves in class 188370.r do not have complex multiplication.Modular form 188370.2.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.
\(\left(\begin{array}{rrrrrrrr} 1 & 3 & 4 & 2 & 12 & 6 & 12 & 4 \\ 3 & 1 & 12 & 6 & 4 & 2 & 4 & 12 \\ 4 & 12 & 1 & 2 & 3 & 6 & 12 & 4 \\ 2 & 6 & 2 & 1 & 6 & 3 & 6 & 2 \\ 12 & 4 & 3 & 6 & 1 & 2 & 4 & 12 \\ 6 & 2 & 6 & 3 & 2 & 1 & 2 & 6 \\ 12 & 4 & 12 & 6 & 4 & 2 & 1 & 3 \\ 4 & 12 & 4 & 2 & 12 & 6 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
