Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2+xy+y=x^3-1217868x+517205302\) | (homogenize, simplify) |
\(y^2z+xyz+yz^2=x^3-1217868xz^2+517205302z^3\) | (dehomogenize, simplify) |
\(y^2=x^3-1578356307x+24135465650670\) | (homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Infinite order Mordell-Weil generator and height
$P$ | = | \(\left(\frac{6283}{9}, \frac{63767}{27}\right)\) |
$\hat{h}(P)$ | ≈ | $1.5143736681599098397935527371$ |
Torsion generators
\( \left(637, -319\right) \)
Integral points
\( \left(187, 17111\right) \), \( \left(187, -17299\right) \), \( \left(637, -319\right) \)
Invariants
Conductor: | \( 726 \) | = | $2 \cdot 3 \cdot 11^{2}$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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Discriminant: | $3423740047332 $ | = | $2^{2} \cdot 3 \cdot 11^{11} $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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j-invariant: | \( \frac{112763292123580561}{1932612} \) | = | $2^{-2} \cdot 3^{-1} \cdot 11^{-5} \cdot 179^{3} \cdot 2699^{3}$ | comment: j-invariant
sage: E.j_invariant().factor()
gp: E.j
magma: jInvariant(E);
oscar: j_invariant(E)
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Endomorphism ring: | $\Z$ | |||
Geometric endomorphism ring: | \(\Z\) | (no potential complex multiplication) | sage: E.has_cm()
magma: HasComplexMultiplication(E);
| |
Sato-Tate group: | $\mathrm{SU}(2)$ | |||
Faltings height: | $1.9468009027957329301462474431\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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Stable Faltings height: | $0.74785326639654765811527565412\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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$abc$ quality: | $1.0637869122465706\dots$ | |||
Szpiro ratio: | $8.144369116382856\dots$ |
BSD invariants
Analytic rank: | $1$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
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Regulator: | $1.5143736681599098397935527371\dots$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
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Real period: | $0.56700993613783485390447295831\dots$ | comment: Real Period
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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Tamagawa product: | $ 8 $ = $ 2\cdot1\cdot2^{2} $ | comment: Tamagawa numbers
sage: E.tamagawa_numbers()
gp: gr=ellglobalred(E); [[gr[4][i,1],gr[5][i][4]] | i<-[1..#gr[4][,1]]]
magma: TamagawaNumbers(E);
oscar: tamagawa_numbers(E)
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Torsion order: | $2$ | comment: Torsion order
sage: E.torsion_order()
gp: elltors(E)[1]
magma: Order(TorsionSubgroup(E));
oscar: prod(torsion_structure(E)[1])
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Analytic order of Ш: | $1$ ( rounded) | comment: Order of Sha
sage: E.sha().an_numerical()
magma: MordellWeilShaInformation(E);
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Special value: | $ L'(E,1) $ ≈ $ 1.7173298337443383786694291511 $ | comment: Special L-value
r = E.rank();
gp: [r,L1r] = ellanalyticrank(E); L1r/r!
magma: Lr1 where r,Lr1 := AnalyticRank(E: Precision:=12);
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BSD formula
$\displaystyle 1.717329834 \approx L'(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.567010 \cdot 1.514374 \cdot 8}{2^2} \approx 1.717329834$
Modular invariants
For more coefficients, see the Downloads section to the right.
Modular degree: | 12000 | comment: Modular degree
sage: E.modular_degree()
gp: ellmoddegree(E)
magma: ModularDegree(E);
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$ \Gamma_0(N) $-optimal: | no | |
Manin constant: | 1 | comment: Manin constant
magma: ManinConstant(E);
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Local data
This elliptic curve is not semistable. There are 3 primes of bad reduction:
prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
---|---|---|---|---|---|---|---|
$2$ | $2$ | $I_{2}$ | Non-split multiplicative | 1 | 1 | 2 | 2 |
$3$ | $1$ | $I_{1}$ | Split multiplicative | -1 | 1 | 1 | 1 |
$11$ | $4$ | $I_{5}^{*}$ | Additive | -1 | 2 | 11 | 5 |
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 |
---|---|---|
$2$ | 2B | 8.6.0.4 |
$5$ | 5B.4.2 | 5.12.0.2 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 1320 = 2^{3} \cdot 3 \cdot 5 \cdot 11 \), index $288$, genus $5$, and generators
$\left(\begin{array}{rr} 1301 & 20 \\ 1300 & 21 \end{array}\right),\left(\begin{array}{rr} 6 & 13 \\ 355 & 1136 \end{array}\right),\left(\begin{array}{rr} 11 & 16 \\ 1080 & 971 \end{array}\right),\left(\begin{array}{rr} 991 & 20 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1057 & 20 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 714 & 1307 \\ 445 & 184 \end{array}\right),\left(\begin{array}{rr} 1 & 10 \\ 10 & 101 \end{array}\right),\left(\begin{array}{rr} 661 & 6 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 20 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 20 \\ 0 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[1320])$ is a degree-$1622016000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/1320\Z)$.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 5 and 10.
Its isogeny class 726e
consists of 4 curves linked by isogenies of
degrees dividing 10.
Twists
The minimal quadratic twist of this elliptic curve is 66c3, its twist by $-11$.
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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z$ are as follows:
$[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
---|---|---|---|
$2$ | \(\Q(\sqrt{33}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | 2.2.33.1-132.1-j7 |
$4$ | 4.0.2112.2 | \(\Z/4\Z\) | Not in database |
$4$ | 4.4.15125.1 | \(\Z/10\Z\) | Not in database |
$8$ | 8.0.4857532416.6 | \(\Z/2\Z \oplus \Z/4\Z\) | Not in database |
$8$ | 8.4.1322463200256.7 | \(\Z/2\Z \oplus \Z/4\Z\) | Not in database |
$8$ | 8.2.5021227463472.3 | \(\Z/6\Z\) | Not in database |
$8$ | 8.8.18530015625.1 | \(\Z/2\Z \oplus \Z/10\Z\) | Not in database |
$10$ | 10.0.2641639027500000000.2 | \(\Z/10\Z\) | Not in database |
$16$ | deg 16 | \(\Z/8\Z\) | Not in database |
$16$ | deg 16 | \(\Z/2\Z \oplus \Z/6\Z\) | Not in database |
$16$ | deg 16 | \(\Z/20\Z\) | Not in database |
$20$ | 20.0.62804310764500311806250000000000000000.1 | \(\Z/2\Z \oplus \Z/10\Z\) | 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.
Iwasawa invariants
$p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Reduction type | nonsplit | split | ord | ord | add | ord | ord | ss | ord | ord | ord | ord | ord | ord | ord |
$\lambda$-invariant(s) | 1 | 4 | 5 | 1 | - | 1 | 1 | 1,1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
$\mu$-invariant(s) | 0 | 0 | 0 | 0 | - | 0 | 0 | 0,0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
An entry - indicates that the invariants are not computed because the reduction is additive.
$p$-adic regulators
$p$-adic regulators are not yet computed for curves that are not $\Gamma_0$-optimal.