Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2+xy+y=x^3-x^2-188x+992\) | (homogenize, simplify) |
\(y^2z+xyz+yz^2=x^3-x^2z-188xz^2+992z^3\) | (dehomogenize, simplify) |
\(y^2=x^3-3003x+60502\) | (homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z \oplus \Z/{2}\Z\)
Infinite order Mordell-Weil generators and heights
$P$ | = | \(\left(10, 1\right)\) | \(\left(-3, 40\right)\) |
$\hat{h}(P)$ | ≈ | $0.90768987742575746838442890795$ | $0.91910416136377897238826981526$ |
Torsion generators
\( \left(\frac{27}{4}, -\frac{31}{8}\right) \)
Integral points
\( \left(-12, 43\right) \), \( \left(-12, -32\right) \), \( \left(-3, 40\right) \), \( \left(-3, -38\right) \), \( \left(0, 31\right) \), \( \left(0, -32\right) \), \( \left(6, 4\right) \), \( \left(6, -11\right) \), \( \left(10, 1\right) \), \( \left(10, -12\right) \), \( \left(13, 18\right) \), \( \left(13, -32\right) \), \( \left(37, 190\right) \), \( \left(37, -228\right) \), \( \left(88, 768\right) \), \( \left(88, -857\right) \), \( \left(556, 12819\right) \), \( \left(556, -13376\right) \)
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: | $37074375 $ | = | $3^{3} \cdot 5^{4} \cdot 13^{3} $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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j-invariant: | \( \frac{12326391}{625} \) | = | $3^{3} \cdot 5^{-4} \cdot 7^{3} \cdot 11^{3}$ | 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: | $0.21065043830241923793638178042\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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Stable Faltings height: | $-0.70523997322999236892580138920\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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$abc$ quality: | $0.9050395539749845\dots$ | |||
Szpiro ratio: | $3.0568736054319428\dots$ |
BSD invariants
Analytic rank: | $2$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
|
Regulator: | $0.82122325689920145708690878515\dots$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
|
Real period: | $2.0285457996202898315598492474\dots$ | 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);
|
Tamagawa product: | $ 8 $ = $ 2\cdot2\cdot2 $ | 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])
|
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! $ ≈ $ 3.3317779766667386358467281443 $ | 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
$\displaystyle 3.331777977 \approx L^{(2)}(E,1)/2! \overset{?}{=} \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 2.028546 \cdot 0.821223 \cdot 8}{2^2} \approx 3.331777977$
Modular invariants
For more coefficients, see the Downloads section to the right.
Modular degree: | 2304 | 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 | $v_p(N)$ | $v_p(\Delta)$ | $v_p(\mathrm{den}(j))$ |
---|---|---|---|---|---|---|---|
$3$ | $2$ | $III$ | additive | 1 | 2 | 3 | 0 |
$5$ | $2$ | $I_{4}$ | nonsplit multiplicative | 1 | 1 | 4 | 4 |
$13$ | $2$ | $III$ | additive | -1 | 2 | 3 | 0 |
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 | 4.6.0.4 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 1560 = 2^{3} \cdot 3 \cdot 5 \cdot 13 \), index $48$, genus $1$, and generators
$\left(\begin{array}{rr} 848 & 3 \\ 317 & 1544 \end{array}\right),\left(\begin{array}{rr} 394 & 393 \\ 21 & 406 \end{array}\right),\left(\begin{array}{rr} 1 & 1178 \\ 1170 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 4 & 17 \end{array}\right),\left(\begin{array}{rr} 6 & 19 \\ 1033 & 1538 \end{array}\right),\left(\begin{array}{rr} 3 & 8 \\ 1552 & 1539 \end{array}\right),\left(\begin{array}{rr} 937 & 8 \\ 628 & 33 \end{array}\right),\left(\begin{array}{rr} 1553 & 8 \\ 1552 & 9 \end{array}\right),\left(\begin{array}{rr} 5 & 8 \\ 48 & 77 \end{array}\right)$.
The torsion field $K:=\Q(E[1560])$ is a degree-$19322634240$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/1560\Z)$.
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 | $2$ | \( 39 = 3 \cdot 13 \) |
$3$ | additive | $6$ | \( 845 = 5 \cdot 13^{2} \) |
$5$ | nonsplit multiplicative | $6$ | \( 1521 = 3^{2} \cdot 13^{2} \) |
$13$ | additive | $50$ | \( 45 = 3^{2} \cdot 5 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2.
Its isogeny class 7605.b
consists of 2 curves linked by isogenies of
degree 2.
Twists
This elliptic curve is its own minimal quadratic twist.
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 |
---|---|---|---|
$2$ | \(\Q(\sqrt{39}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
$4$ | 4.0.237276.1 | \(\Z/4\Z\) | not in database |
$8$ | 8.4.230604391120896.18 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
$8$ | 8.0.900798402816.3 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
$8$ | 8.2.6597644551875.6 | \(\Z/6\Z\) | not in database |
$16$ | deg 16 | \(\Z/4\Z \oplus \Z/4\Z\) | 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 |
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 | nonsplit | ord | ss | add | ord | ord | ord | ord | ord | ord | ord | ord | ord |
$\lambda$-invariant(s) | 3 | - | 2 | 2 | 2,2 | - | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
$\mu$-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.