Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2+xy+y=x^3+x^2-875x-8440\)
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(homogenize, simplify) |
\(y^2z+xyz+yz^2=x^3+x^2z-875xz^2-8440z^3\)
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(dehomogenize, simplify) |
\(y^2=x^3-1134027x-376757946\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z \oplus \Z/{2}\Z\)
Infinite order Mordell-Weil generator and height
$P$ | = |
\(\left(-22, 38\right)\)
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$\hat{h}(P)$ | ≈ | $0.85009030288104411292552745618$ |
Torsion generators
\( \left(-11, 5\right) \), \( \left(33, -17\right) \)
Integral points
\( \left(-22, 38\right) \), \( \left(-22, -17\right) \), \( \left(-12, 28\right) \), \( \left(-12, -17\right) \), \( \left(-11, 5\right) \), \( \left(33, -17\right) \), \( \left(38, 103\right) \), \( \left(38, -142\right) \), \( \left(133, 1433\right) \), \( \left(133, -1567\right) \), \( \left(528, 11863\right) \), \( \left(528, -12392\right) \)
Invariants
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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Conductor: | \( 1155 \) | = | $3 \cdot 5 \cdot 7 \cdot 11$ |
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
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Discriminant: | $14707625625 $ | = | $3^{4} \cdot 5^{4} \cdot 7^{4} \cdot 11^{2} $ |
sage: E.j_invariant().factor()
gp: E.j
magma: jInvariant(E);
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j-invariant: | \( \frac{74093292126001}{14707625625} \) | = | $3^{-4} \cdot 5^{-4} \cdot 7^{-4} \cdot 11^{-2} \cdot 97^{3} \cdot 433^{3}$ |
Endomorphism ring: | $\Z$ | ||
Geometric endomorphism ring: | \(\Z\) | (no potential complex multiplication) | |
Sato-Tate group: | $\mathrm{SU}(2)$ | ||
Faltings height: | $0.66729310237448382314064497860\dots$ | ||
Stable Faltings height: | $0.66729310237448382314064497860\dots$ |
BSD invariants
sage: E.rank()
magma: Rank(E);
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Analytic rank: | $1$ | ||
sage: E.regulator()
magma: Regulator(E);
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Regulator: | $0.85009030288104411292552745618\dots$ | ||
sage: E.period_lattice().omega()
gp: if(E.disc>0,2,1)*E.omega[1]
magma: (Discriminant(E) gt 0 select 2 else 1) * RealPeriod(E);
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Real period: | $0.88997304932837026230710109381\dots$ | ||
sage: E.tamagawa_numbers()
gp: gr=ellglobalred(E); [[gr[4][i,1],gr[5][i][4]] | i<-[1..#gr[4][,1]]]
magma: TamagawaNumbers(E);
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Tamagawa product: | $ 32 $ = $ 2\cdot2^{2}\cdot2\cdot2 $ | ||
sage: E.torsion_order()
gp: elltors(E)[1]
magma: Order(TorsionSubgroup(E));
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Torsion order: | $4$ | ||
sage: E.sha().an_numerical()
magma: MordellWeilShaInformation(E);
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Analytic order of Ш: | $1$ (exact) | ||
sage: r = E.rank();
gp: ar = ellanalyticrank(E);
magma: Lr1 where r,Lr1 := AnalyticRank(E: Precision:=12);
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Special value: | $ L'(E,1) $ ≈ $ 1.5131149181190413784512069043 $ |
Modular invariants
For more coefficients, see the Downloads section to the right.
sage: E.modular_degree()
magma: ModularDegree(E);
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Modular degree: | 1024 | ||
$ \Gamma_0(N) $-optimal: | no | ||
Manin constant: | 1 |
Local data
This elliptic curve is semistable. There are 4 primes of bad reduction:
prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
---|---|---|---|---|---|---|---|
$3$ | $2$ | $I_{4}$ | Non-split multiplicative | 1 | 1 | 4 | 4 |
$5$ | $4$ | $I_{4}$ | Split multiplicative | -1 | 1 | 4 | 4 |
$7$ | $2$ | $I_{4}$ | Non-split multiplicative | 1 | 1 | 4 | 4 |
$11$ | $2$ | $I_{2}$ | Split multiplicative | -1 | 1 | 2 | 2 |
Galois representations
The $\ell$-adic Galois representation has maximal image for all primes $\ell$ except those listed in the table below.
prime $\ell$ | mod-$\ell$ image | $\ell$-adic image |
---|---|---|
$2$ | 2Cs | 4.24.0.6 |
The image of the adelic Galois representation has level $9240$, index $192$, genus $1$, and generators
$\left(\begin{array}{rr} 9233 & 8 \\ 9232 & 9 \end{array}\right),\left(\begin{array}{rr} 8407 & 2 \\ 822 & 9235 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 6161 & 8 \\ 6164 & 33 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 5281 & 4 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 5 & 4 \\ 9236 & 9237 \end{array}\right),\left(\begin{array}{rr} 4625 & 6932 \\ 6952 & 4629 \end{array}\right),\left(\begin{array}{rr} 7 & 6936 \\ 9234 & 2305 \end{array}\right),\left(\begin{array}{rr} 3697 & 8 \\ 5548 & 33 \end{array}\right)$
$p$-adic regulators
$p$-adic regulators are not yet computed for curves that are not $\Gamma_0$-optimal.
Iwasawa invariants
$p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 |
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Reduction type | ord | nonsplit | split | nonsplit | split | ord | ord | ord | ord | ord | ord | ord | ord | ord | ord |
$\lambda$-invariant(s) | 1 | 1 | 2 | 3 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
$\mu$-invariant(s) | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2 and 4.
Its isogeny class 1155.e
consists of 6 curves linked by isogenies of
degrees dividing 8.
Growth of torsion in number fields
The number fields $K$ of degree less than 24 such that $E(K)_{\rm tors}$ is strictly larger than $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z \oplus \Z/{2}\Z$ are as follows:
$[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
---|---|---|---|
$2$ | \(\Q(\sqrt{-11}) \) | \(\Z/2\Z \oplus \Z/4\Z\) | Not in database |
$2$ | \(\Q(\sqrt{11}) \) | \(\Z/2\Z \oplus \Z/4\Z\) | Not in database |
$2$ | \(\Q(\sqrt{-1}) \) | \(\Z/2\Z \oplus \Z/4\Z\) | Not in database |
$4$ | \(\Q(i, \sqrt{11})\) | \(\Z/4\Z \oplus \Z/4\Z\) | Not in database |
$8$ | 8.0.1088900598016.3 | \(\Z/2\Z \oplus \Z/8\Z\) | Not in database |
$8$ | 8.4.5877614223360000.10 | \(\Z/2\Z \oplus \Z/8\Z\) | Not in database |
$8$ | 8.0.7965941760000.3 | \(\Z/2\Z \oplus \Z/8\Z\) | Not in database |
$8$ | deg 8 | \(\Z/2\Z \oplus \Z/8\Z\) | Not in database |
$8$ | 8.2.3892034846266875.7 | \(\Z/2\Z \oplus \Z/6\Z\) | Not in database |
$16$ | deg 16 | \(\Z/4\Z \oplus \Z/8\Z\) | Not in database |
$16$ | deg 16 | \(\Z/4\Z \oplus \Z/8\Z\) | Not in database |
$16$ | deg 16 | \(\Z/4\Z \oplus \Z/8\Z\) | Not in database |
$16$ | deg 16 | \(\Z/2\Z \oplus \Z/12\Z\) | Not in database |
$16$ | deg 16 | \(\Z/2\Z \oplus \Z/12\Z\) | Not in database |
$16$ | deg 16 | \(\Z/2\Z \oplus \Z/12\Z\) | Not in database |
We only show fields where the torsion growth is primitive. For fields not in the database, click on the degree shown to reveal the defining polynomial.