Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2+xy+y=x^3-x^2-790952x+248249954\) | (homogenize, simplify) |
\(y^2z+xyz+yz^2=x^3-x^2z-790952xz^2+248249954z^3\) | (dehomogenize, simplify) |
\(y^2=x^3-12655227x+15875341846\) | (homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z \oplus \Z/{2}\Z\)
Infinite order Mordell-Weil generator and height
$P$ | = | \(\left(894, 15523\right)\) |
$\hat{h}(P)$ | ≈ | $3.1817406120739643590554055418$ |
Torsion generators
\( \left(-1017, 508\right) \), \( \left(387, -194\right) \)
Integral points
\( \left(-1017, 508\right) \), \( \left(-288, 21406\right) \), \( \left(-288, -21119\right) \), \( \left(387, -194\right) \), \( \left(894, 15523\right) \), \( \left(894, -16418\right) \)
Invariants
Conductor: | \( 7605 \) | = | $3^{2} \cdot 5 \cdot 13^{2}$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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Discriminant: | $5087753803244750625 $ | = | $3^{10} \cdot 5^{4} \cdot 13^{10} $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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j-invariant: | \( \frac{15551989015681}{1445900625} \) | = | $3^{-4} \cdot 5^{-4} \cdot 13^{-4} \cdot 109^{3} \cdot 229^{3}$ | 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: | $2.3280507102995117912813310913\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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Stable Faltings height: | $0.49626988723468857755696475205\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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$abc$ quality: | $0.9738373897310543\dots$ | |||
Szpiro ratio: | $5.858694226435877\dots$ |
BSD invariants
Analytic rank: | $1$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
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Regulator: | $3.1817406120739643590554055418\dots$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
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Real period: | $0.23604752447090842518772862862\dots$ | 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: | $ 64 $ = $ 2^{2}\cdot2^{2}\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: | $4$ | 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) $ ≈ $ 3.0041679799544490112274631941 $ | 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
$\displaystyle 3.004167980 \approx L'(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.236048 \cdot 3.181741 \cdot 64}{4^2} \approx 3.004167980$
Modular invariants
For more coefficients, see the Downloads section to the right.
Modular degree: | 129024 | 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 $p$ of bad reduction:
$p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $v_p(N)$ | $v_p(\Delta)$ | $v_p(\mathrm{den}(j))$ |
---|---|---|---|---|---|---|---|
$3$ | $4$ | $I_{4}^{*}$ | additive | -1 | 2 | 10 | 4 |
$5$ | $4$ | $I_{4}$ | split multiplicative | -1 | 1 | 4 | 4 |
$13$ | $4$ | $I_{4}^{*}$ | additive | 1 | 2 | 10 | 4 |
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 | 8.48.0.48 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 3120 = 2^{4} \cdot 3 \cdot 5 \cdot 13 \), index $768$, genus $13$, and generators
$\left(\begin{array}{rr} 1 & 16 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 16 \\ 8 & 129 \end{array}\right),\left(\begin{array}{rr} 2741 & 16 \\ 3056 & 1857 \end{array}\right),\left(\begin{array}{rr} 2395 & 3116 \\ 764 & 35 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 16 & 1 \end{array}\right),\left(\begin{array}{rr} 633 & 16 \\ 188 & 2345 \end{array}\right),\left(\begin{array}{rr} 1039 & 3108 \\ 0 & 3119 \end{array}\right),\left(\begin{array}{rr} 1959 & 28 \\ 1316 & 2361 \end{array}\right),\left(\begin{array}{rr} 3105 & 16 \\ 3104 & 17 \end{array}\right)$.
The torsion field $K:=\Q(E[3120])$ is a degree-$19322634240$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/3120\Z)$.
The table below list all primes $\ell$ for which the Serre invariants associated to the mod-$\ell$ Galois representation are exceptional.
$\ell$ | Reduction type | Serre weight | Serre conductor |
---|---|---|---|
$2$ | good | $2$ | \( 1521 = 3^{2} \cdot 13^{2} \) |
$3$ | additive | $8$ | \( 845 = 5 \cdot 13^{2} \) |
$5$ | split multiplicative | $6$ | \( 1521 = 3^{2} \cdot 13^{2} \) |
$13$ | additive | $98$ | \( 45 = 3^{2} \cdot 5 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2 and 4.
Its isogeny class 7605.h
consists of 8 curves linked by isogenies of
degrees dividing 16.
Twists
The minimal quadratic twist of this elliptic curve is 195.a4, its twist by $-39$.
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 |
---|---|---|---|
$2$ | \(\Q(\sqrt{39}) \) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
$2$ | \(\Q(\sqrt{-39}) \) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
$4$ | \(\Q(i, \sqrt{39})\) | \(\Z/4\Z \oplus \Z/4\Z\) | not in database |
$4$ | \(\Q(\sqrt{-3}, \sqrt{13})\) | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
$8$ | 8.8.94758543360000.5 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
$8$ | 8.0.94758543360000.123 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
$8$ | 8.0.592240896.1 | \(\Z/4\Z \oplus \Z/8\Z\) | not in database |
$8$ | 8.2.534409208701875.1 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
$16$ | 16.0.8979181539709000089600000000.26 | \(\Z/4\Z \oplus \Z/8\Z\) | not in database |
$16$ | 16.0.8979181539709000089600000000.4 | \(\Z/4\Z \oplus \Z/8\Z\) | not in database |
$16$ | deg 16 | \(\Z/2\Z \oplus \Z/16\Z\) | 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.
Iwasawa invariants
$p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Reduction type | ord | add | split | ss | ord | add | ord | ord | ord | ord | ord | ord | ord | ord | ord |
$\lambda$-invariant(s) | 3 | - | 2 | 1,3 | 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 |
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.