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SageMath
E = EllipticCurve("h1")
E.isogeny_class()
Elliptic curves in class 313950.h
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 313950.h1 | 313950h7 | \([1, 1, 0, -1076444963750, 429803647574100000]\) | \(8828342566147309471108534663879471201/1536341563898865415843582949700\) | \(24005336935919772122555983589062500\) | \([2]\) | \(5478285312\) | \(5.6025\) | |
| 313950.h2 | 313950h4 | \([1, 1, 0, -1076400001250, 429841353250312500]\) | \(8827236347661221188886967161105287201/46539238473000000\) | \(727175601140625000000\) | \([2]\) | \(1826095104\) | \(5.0532\) | |
| 313950.h3 | 313950h8 | \([1, 1, 0, -461748530750, -116725780133631000]\) | \(696819431300451649932999125896765921/26551778102890598266349123437500\) | \(414871532857665597911705053710937500\) | \([2]\) | \(5478285312\) | \(5.6025\) | |
| 313950.h4 | 313950h6 | \([1, 1, 0, -74036251250, 5284560239062500]\) | \(2872347286043717137884962530087201/890114999660918118510786090000\) | \(13908046869701845601731032656250000\) | \([2, 2]\) | \(2739142656\) | \(5.2560\) | |
| 313950.h5 | 313950h5 | \([1, 1, 0, -67412593250, 6687397796056500]\) | \(2168337038351679228688694521765921/18360082088470458984375000000\) | \(286876282632350921630859375000000\) | \([2]\) | \(1826095104\) | \(5.0532\) | |
| 313950.h6 | 313950h2 | \([1, 1, 0, -67275001250, 6716249875312500]\) | \(2155087111607167363355460545287201/156481162929000000000000\) | \(2445018170765625000000000000\) | \([2, 2]\) | \(913047552\) | \(4.7067\) | |
| 313950.h7 | 313950h1 | \([1, 1, 0, -4196089250, 105390660976500]\) | \(-522923112164227281987660878881/4484275679769919488000000\) | \(-70066807496404992000000000000\) | \([2]\) | \(456523776\) | \(4.3601\) | \(\Gamma_0(N)\)-optimal |
| 313950.h8 | 313950h3 | \([1, 1, 0, 12846310750, 557888218576500]\) | \(15005102139088880168192111025119/17288486242801155608083603200\) | \(-270132597543768056376306300000000\) | \([2]\) | \(1369571328\) | \(4.9094\) |
Rank
sage: E.rank()
The elliptic curves in class 313950.h have rank \(0\).
Complex multiplication
The elliptic curves in class 313950.h do not have complex multiplication.Modular form 313950.2.a.h
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.
