Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2+xy+y=x^3-x^2-7017908x-7154068534\)
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(homogenize, simplify) |
\(y^2z+xyz+yz^2=x^3-x^2z-7017908xz^2-7154068534z^3\)
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(dehomogenize, simplify) |
\(y^2=x^3-112286523x-457972672682\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Infinite order Mordell-Weil generator and height
$P$ | = |
\(\left(\frac{241989799}{13924}, \frac{3703451766397}{1643032}\right)\)
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$\hat{h}(P)$ | ≈ | $16.444300196502978986772768780$ |
Torsion generators
\( \left(-\frac{6117}{4}, \frac{6113}{8}\right) \)
Integral points
None
Invariants
Conductor: | \( 16245 \) | = | $3^{2} \cdot 5 \cdot 19^{2}$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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Discriminant: | $13890061135845 $ | = | $3^{10} \cdot 5 \cdot 19^{6} $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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j-invariant: | \( \frac{1114544804970241}{405} \) | = | $3^{-4} \cdot 5^{-1} \cdot 103681^{3}$ | 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);
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Sato-Tate group: | $\mathrm{SU}(2)$ | |||
Faltings height: | $2.3123952423943765976757719587\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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Stable Faltings height: | $0.29086960847710152197363562429\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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$abc$ quality: | $1.0735374703334082\dots$ | |||
Szpiro ratio: | $6.075528269116273\dots$ |
BSD invariants
Analytic rank: | $1$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
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Regulator: | $16.444300196502978986772768780\dots$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
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Real period: | $0.092757202448663634451023939675\dots$ | 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) $ ≈ $ 3.0506545649072520152891014913 $ | 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
$\displaystyle 3.050654565 \approx L'(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.092757 \cdot 16.444300 \cdot 8}{2^2} \approx 3.050654565$
Modular invariants
For more coefficients, see the Downloads section to the right.
Modular degree: | 221184 | 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 | $v_p(N)$ | $v_p(\Delta)$ | $v_p(\mathrm{den}(j))$ |
---|---|---|---|---|---|---|---|
$3$ | $2$ | $I_{4}^{*}$ | additive | -1 | 2 | 10 | 4 |
$5$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
$19$ | $4$ | $I_0^{*}$ | additive | -1 | 2 | 6 | 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 | 16.48.0.121 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 9120 = 2^{5} \cdot 3 \cdot 5 \cdot 19 \), index $768$, genus $13$, and generators
$\left(\begin{array}{rr} 23 & 18 \\ 6558 & 7115 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 32 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 32 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 5 & 28 \\ 68 & 381 \end{array}\right),\left(\begin{array}{rr} 5701 & 2432 \\ 2356 & 2433 \end{array}\right),\left(\begin{array}{rr} 7199 & 0 \\ 0 & 9119 \end{array}\right),\left(\begin{array}{rr} 9089 & 32 \\ 9088 & 33 \end{array}\right),\left(\begin{array}{rr} 6271 & 2432 \\ 8550 & 1 \end{array}\right),\left(\begin{array}{rr} 5872 & 6745 \\ 1463 & 7506 \end{array}\right),\left(\begin{array}{rr} 8777 & 4294 \\ 8778 & 949 \end{array}\right)$.
The torsion field $K:=\Q(E[9120])$ is a degree-$1452382617600$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/9120\Z)$.
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 |
---|---|---|---|
$3$ | additive | $8$ | \( 1805 = 5 \cdot 19^{2} \) |
$5$ | nonsplit multiplicative | $6$ | \( 3249 = 3^{2} \cdot 19^{2} \) |
$19$ | additive | $182$ | \( 45 = 3^{2} \cdot 5 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 4, 8 and 16.
Its isogeny class 16245.c
consists of 8 curves linked by isogenies of
degrees dividing 16.
Twists
The minimal quadratic twist of this elliptic curve is 15.a1, its twist by $57$.
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 |
---|---|---|---|
$2$ | \(\Q(\sqrt{5}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
$2$ | \(\Q(\sqrt{-285}) \) | \(\Z/4\Z\) | not in database |
$2$ | \(\Q(\sqrt{-57}) \) | \(\Z/4\Z\) | not in database |
$4$ | \(\Q(\sqrt{5}, \sqrt{-57})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
$4$ | \(\Q(\sqrt{2}, \sqrt{-57})\) | \(\Z/8\Z\) | not in database |
$4$ | \(\Q(\sqrt{10}, \sqrt{-57})\) | \(\Z/8\Z\) | not in database |
$8$ | 8.4.42224004000000.33 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
$8$ | 8.0.10809345024000000.207 | \(\Z/8\Z\) | not in database |
$8$ | 8.0.432373800960000.208 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
$8$ | 8.0.2767192326144000000.9 | \(\Z/16\Z\) | not in database |
$8$ | 8.0.2767192326144000000.18 | \(\Z/16\Z\) | not in database |
$8$ | 8.2.14428733866875.1 | \(\Z/6\Z\) | not in database |
$16$ | deg 16 | \(\Z/4\Z \oplus \Z/4\Z\) | not in database |
$16$ | deg 16 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
$16$ | deg 16 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
$16$ | deg 16 | \(\Z/16\Z\) | not in database |
$16$ | 16.0.7657353369870241665908736000000000000.67 | \(\Z/2\Z \oplus \Z/16\Z\) | not in database |
$16$ | deg 16 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
$16$ | deg 16 | \(\Z/12\Z\) | not in database |
$16$ | deg 16 | \(\Z/12\Z\) | 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 | nonsplit | ss | ord | ord | ord | add | ss | ord | ss | ord | ord | ord | ord |
$\lambda$-invariant(s) | 3 | - | 3 | 1,5 | 1 | 1 | 1 | - | 1,3 | 1 | 1,1 | 1 | 1 | 1 | 1 |
$\mu$-invariant(s) | 3 | - | 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.