Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2=x^3-x^2-2977x+63521\) | (homogenize, simplify) |
\(y^2z=x^3-x^2z-2977xz^2+63521z^3\) | (dehomogenize, simplify) |
\(y^2=x^3-241164x+45583344\) | (homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z\)
Infinite order Mordell-Weil generators and heights
$P$ | = | \(\left(25, 64\right)\) | \(\left(31, 8\right)\) |
$\hat{h}(P)$ | ≈ | $0.40569193325419347689320435183$ | $1.2741630005422245239517563437$ |
Integral points
\((-13,\pm 316)\), \((19,\pm 116)\), \((25,\pm 64)\), \((31,\pm 8)\), \((32,\pm 1)\), \((89,\pm 704)\), \((857,\pm 25024)\)
Invariants
Conductor: | \( 5056 \) | = | $2^{6} \cdot 79$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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Discriminant: | $82837504 $ | = | $2^{20} \cdot 79 $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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j-invariant: | \( \frac{11134383337}{316} \) | = | $2^{-2} \cdot 7^{3} \cdot 11^{3} \cdot 29^{3} \cdot 79^{-1}$ | 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: | $0.62096655155359358720382325763\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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Stable Faltings height: | $-0.41875421928632437692202492456\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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BSD invariants
Analytic rank: | $2$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
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Regulator: | $0.50656766238066061222731896129\dots$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
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Real period: | $1.7875239462134869075808023436\dots$ | 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: | $ 4 $ = $ 2^{2}\cdot1 $ | 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: | $1$ | 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^{(2)}(E,1)/2! $ ≈ $ 3.6220073075312791015097448597 $ | 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
$\displaystyle 3.622007308 \approx L^{(2)}(E,1)/2! \overset{?}{=} \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 1.787524 \cdot 0.506568 \cdot 4}{1^2} \approx 3.622007308$
Modular invariants
For more coefficients, see the Downloads section to the right.
Modular degree: | 2560 | comment: Modular degree
sage: E.modular_degree()
gp: ellmoddegree(E)
magma: ModularDegree(E);
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$ \Gamma_0(N) $-optimal: | yes | |
Manin constant: | 1 | comment: Manin constant
magma: ManinConstant(E);
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Local data
This elliptic curve is not semistable. There are 2 primes of bad reduction:
prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
---|---|---|---|---|---|---|---|
$2$ | $4$ | $I_{10}^{*}$ | Additive | 1 | 6 | 20 | 2 |
$79$ | $1$ | $I_{1}$ | Split multiplicative | -1 | 1 | 1 | 1 |
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 |
---|---|---|
$3$ | 3B | 9.12.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 5688 = 2^{3} \cdot 3^{2} \cdot 79 \), index $144$, genus $3$, and generators
$\left(\begin{array}{rr} 1 & 18 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1423 & 18 \\ 1431 & 163 \end{array}\right),\left(\begin{array}{rr} 2843 & 0 \\ 0 & 5687 \end{array}\right),\left(\begin{array}{rr} 4255 & 5670 \\ 3006 & 2951 \end{array}\right),\left(\begin{array}{rr} 1 & 18 \\ 10 & 181 \end{array}\right),\left(\begin{array}{rr} 10 & 9 \\ 81 & 73 \end{array}\right),\left(\begin{array}{rr} 5671 & 18 \\ 5670 & 19 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 18 & 1 \end{array}\right),\left(\begin{array}{rr} 3911 & 5670 \\ 3159 & 5177 \end{array}\right),\left(\begin{array}{rr} 2861 & 5670 \\ 2862 & 5669 \end{array}\right)$.
The torsion field $K:=\Q(E[5688])$ is a degree-$1594634895360$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/5688\Z)$.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
3 and 9.
Its isogeny class 5056.d
consists of 3 curves linked by isogenies of
degrees dividing 9.
Twists
The minimal quadratic twist of this elliptic curve is 158.b3, its twist by $8$.
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 $E(\Q)_{\rm tors}$ (which is trivial) are as follows:
$[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
---|---|---|---|
$2$ | \(\Q(\sqrt{2}) \) | \(\Z/3\Z\) | Not in database |
$3$ | 3.3.316.1 | \(\Z/2\Z\) | Not in database |
$6$ | 6.6.31554496.1 | \(\Z/2\Z \oplus \Z/2\Z\) | Not in database |
$6$ | 6.0.2153783678976.6 | \(\Z/3\Z\) | Not in database |
$6$ | 6.6.19942441472.1 | \(\Z/9\Z\) | Not in database |
$6$ | 6.6.12781568.1 | \(\Z/6\Z\) | Not in database |
$12$ | deg 12 | \(\Z/4\Z\) | Not in database |
$12$ | deg 12 | \(\Z/3\Z \oplus \Z/3\Z\) | Not in database |
$12$ | deg 12 | \(\Z/9\Z\) | Not in database |
$12$ | deg 12 | \(\Z/2\Z \oplus \Z/6\Z\) | Not in database |
$18$ | 18.0.997655061193989109767028639057038494662656.1 | \(\Z/6\Z\) | Not in database |
$18$ | 18.18.3167883011970955614511979370068836352.1 | \(\Z/18\Z\) | 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 | 79 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Reduction type | add | ord | ord | ord | ss | ord | ss | ord | ord | ss | ord | ord | ord | ord | ord | split |
$\lambda$-invariant(s) | - | 2 | 2 | 2 | 2,2 | 2 | 2,2 | 2 | 2 | 2,2 | 2 | 2 | 2 | 2 | 2 | 3 |
$\mu$-invariant(s) | - | 0 | 0 | 0 | 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
Note: $p$-adic regulator data only exists for primes $p\ge 5$ of good ordinary reduction.