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SageMath
E = EllipticCurve("fi1")
E.isogeny_class()
Elliptic curves in class 137280.fi
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
137280.fi1 | 137280bs8 | \([0, 1, 0, -67008001, 118677373439]\) | \(126929854754212758768001/50235797102795981820\) | \(13169012795715349858222080\) | \([2]\) | \(31850496\) | \(3.5184\) | |
137280.fi2 | 137280bs6 | \([0, 1, 0, -58489601, 172099667199]\) | \(84415028961834287121601/30783551683856400\) | \(8069723372612852121600\) | \([2, 2]\) | \(15925248\) | \(3.1718\) | |
137280.fi3 | 137280bs3 | \([0, 1, 0, -58484481, 172131318015]\) | \(84392862605474684114881/11228954880\) | \(2943603148062720\) | \([2]\) | \(7962624\) | \(2.8252\) | |
137280.fi4 | 137280bs7 | \([0, 1, 0, -50053121, 223496390655]\) | \(-52902632853833942200321/51713453577420277500\) | \(-13556371574599261224960000\) | \([2]\) | \(31850496\) | \(3.5184\) | |
137280.fi5 | 137280bs5 | \([0, 1, 0, -30201601, -63887865601]\) | \(11621808143080380273601/1335706803288000\) | \(350147524241129472000\) | \([2]\) | \(10616832\) | \(2.9691\) | |
137280.fi6 | 137280bs2 | \([0, 1, 0, -2041601, -826361601]\) | \(3590017885052913601/954068544000000\) | \(250103344398336000000\) | \([2, 2]\) | \(5308416\) | \(2.6225\) | |
137280.fi7 | 137280bs1 | \([0, 1, 0, -730881, 229816575]\) | \(164711681450297281/8097103872000\) | \(2122607197421568000\) | \([2]\) | \(2654208\) | \(2.2759\) | \(\Gamma_0(N)\)-optimal |
137280.fi8 | 137280bs4 | \([0, 1, 0, 5146879, -5339289345]\) | \(57519563401957999679/80296734375000000\) | \(-21049307136000000000000\) | \([2]\) | \(10616832\) | \(2.9691\) |
Rank
sage: E.rank()
The elliptic curves in class 137280.fi have rank \(1\).
Complex multiplication
The elliptic curves in class 137280.fi do not have complex multiplication.Modular form 137280.2.a.fi
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 4 & 3 & 6 & 12 & 12 \\ 2 & 1 & 2 & 2 & 6 & 3 & 6 & 6 \\ 4 & 2 & 1 & 4 & 12 & 6 & 3 & 12 \\ 4 & 2 & 4 & 1 & 12 & 6 & 12 & 3 \\ 3 & 6 & 12 & 12 & 1 & 2 & 4 & 4 \\ 6 & 3 & 6 & 6 & 2 & 1 & 2 & 2 \\ 12 & 6 & 3 & 12 & 4 & 2 & 1 & 4 \\ 12 & 6 & 12 & 3 & 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.