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SageMath
E = EllipticCurve("hc1")
E.isogeny_class()
Elliptic curves in class 450450.hc
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
450450.hc1 | 450450hc8 | \([1, -1, 1, -43875897980, -3529494039626103]\) | \(820076206880893214178646273009/2122496008872985839843750\) | \(24176556101068854331970214843750\) | \([2]\) | \(1783627776\) | \(4.8977\) | |
450450.hc2 | 450450hc5 | \([1, -1, 1, -43848607730, -3534113385656103]\) | \(818546927584539194367471866449/14273634375000\) | \(162585616552734375000\) | \([2]\) | \(594542592\) | \(4.3484\) | |
450450.hc3 | 450450hc7 | \([1, -1, 1, -39990908480, 3065791733631897]\) | \(620954771108295351491118574129/2882378618771462717156250\) | \(32832093954443692512607910156250\) | \([2]\) | \(1783627776\) | \(4.8977\) | \(\Gamma_0(N)\)-optimal* |
450450.hc4 | 450450hc6 | \([1, -1, 1, -3815127230, -7992047618103]\) | \(539142086340577084766074129/309580507925165039062500\) | \(3526315473085083023071289062500\) | \([2, 2]\) | \(891813888\) | \(4.5511\) | \(\Gamma_0(N)\)-optimal* |
450450.hc5 | 450450hc4 | \([1, -1, 1, -2783065730, -53417682080103]\) | \(209289070072300727183442769/12893854589717635333800\) | \(146869062436002439974065625000\) | \([2]\) | \(594542592\) | \(4.3484\) | \(\Gamma_0(N)\)-optimal* |
450450.hc6 | 450450hc2 | \([1, -1, 1, -2740540730, -55219891580103]\) | \(199841159336796255944706769/834505270358760000\) | \(9505536595180250625000000\) | \([2, 2]\) | \(297271296\) | \(4.0018\) | \(\Gamma_0(N)\)-optimal* |
450450.hc7 | 450450hc1 | \([1, -1, 1, -168628730, -890822492103]\) | \(-46555485820017544148689/3157693080314572800\) | \(-35968097742958180800000000\) | \([2]\) | \(148635648\) | \(3.6552\) | \(\Gamma_0(N)\)-optimal* |
450450.hc8 | 450450hc3 | \([1, -1, 1, 949657270, -997343972103]\) | \(8315279469612171276463151/4849789796887785750000\) | \(-55242136905174934558593750000\) | \([2]\) | \(445906944\) | \(4.2046\) | \(\Gamma_0(N)\)-optimal* |
Rank
sage: E.rank()
The elliptic curves in class 450450.hc have rank \(0\).
Complex multiplication
The elliptic curves in class 450450.hc do not have complex multiplication.Modular form 450450.2.a.hc
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.