Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-x^2-21009496x-45319243280\)
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(homogenize, simplify) |
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\(y^2z=x^3-x^2z-21009496xz^2-45319243280z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-1701769203x-33042833658702\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(115592130489622756839435648069091521/8264096906103412850778207779344, 36782374176125620113164933624663250397952119779597319/23757082274654825811904947991072174192257547328)$ | $80.915224871646780272969221927$ | $\infty$ |
| $(5420, 0)$ | $0$ | $2$ |
Integral points
\( \left(5420, 0\right) \)
Invariants
| Conductor: | $N$ | = | \( 96720 \) | = | $2^{4} \cdot 3 \cdot 5 \cdot 13 \cdot 31$ |
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| Discriminant: | $\Delta$ | = | $-294022005272510201856000$ | = | $-1 \cdot 2^{56} \cdot 3^{4} \cdot 5^{3} \cdot 13 \cdot 31 $ |
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| j-invariant: | $j$ | = | \( -\frac{250386371942892200094169}{71782716130983936000} \) | = | $-1 \cdot 2^{-44} \cdot 3^{-4} \cdot 5^{-3} \cdot 13^{-1} \cdot 31^{-1} \cdot 241^{3} \cdot 261529^{3}$ |
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| Endomorphism ring: | $\mathrm{End}(E)$ | = | $\Z$ | |||
| Geometric endomorphism ring: | $\mathrm{End}(E_{\overline{\Q}})$ | = | \(\Z\) (no potential complex multiplication) |
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| Sato-Tate group: | $\mathrm{ST}(E)$ | = | $\mathrm{SU}(2)$ | |||
| Faltings height: | $h_{\mathrm{Faltings}}$ | ≈ | $3.2182040803259142914750805430$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $2.5250568997659689820578484215$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9959744351463596$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $5.452941165341474$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $80.915224871646780272969221927$ |
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| Real period: | $\Omega$ | ≈ | $0.034743196229799797884012875977$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 8 $ = $ 2^{2}\cdot2\cdot1\cdot1\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $5.6225070713880025020731699819 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 5.622507071 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.034743 \cdot 80.915225 \cdot 8}{2^2} \\ & \approx 5.622507071\end{aligned}$$
Modular invariants
Modular form 96720.2.a.o
For more coefficients, see the Downloads section to the right.
| Modular degree: | 11354112 |
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| $ \Gamma_0(N) $-optimal: | yes | |
| Manin constant: | 1 |
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Local data at primes of bad reduction
This elliptic curve is not semistable. There are 5 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $4$ | $I_{48}^{*}$ | additive | -1 | 4 | 56 | 44 |
| $3$ | $2$ | $I_{4}$ | nonsplit multiplicative | 1 | 1 | 4 | 4 |
| $5$ | $1$ | $I_{3}$ | nonsplit multiplicative | 1 | 1 | 3 | 3 |
| $13$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
| $31$ | $1$ | $I_{1}$ | 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 |
|---|---|---|
| $2$ | 2B | 8.12.0.12 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 16120 = 2^{3} \cdot 5 \cdot 13 \cdot 31 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 7 & 6 \\ 16114 & 16115 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 16113 & 8 \\ 16112 & 9 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 4 & 17 \end{array}\right),\left(\begin{array}{rr} 5728 & 3 \\ 1565 & 2 \end{array}\right),\left(\begin{array}{rr} 10071 & 10072 \\ 2002 & 10065 \end{array}\right),\left(\begin{array}{rr} 3228 & 1 \\ 6471 & 6 \end{array}\right),\left(\begin{array}{rr} 14888 & 3 \\ 1245 & 2 \end{array}\right),\left(\begin{array}{rr} 10083 & 10078 \\ 6050 & 14107 \end{array}\right)$.
The torsion field $K:=\Q(E[16120])$ is a degree-$359400996864000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/16120\Z)$.
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 |
|---|---|---|---|
| $2$ | additive | $2$ | \( 2015 = 5 \cdot 13 \cdot 31 \) |
| $3$ | nonsplit multiplicative | $4$ | \( 6448 = 2^{4} \cdot 13 \cdot 31 \) |
| $5$ | nonsplit multiplicative | $6$ | \( 19344 = 2^{4} \cdot 3 \cdot 13 \cdot 31 \) |
| $13$ | nonsplit multiplicative | $14$ | \( 7440 = 2^{4} \cdot 3 \cdot 5 \cdot 31 \) |
| $31$ | split multiplicative | $32$ | \( 3120 = 2^{4} \cdot 3 \cdot 5 \cdot 13 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2 and 4.
Its isogeny class 96720.o
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
The minimal quadratic twist of this elliptic curve is 12090.k4, its twist by $-4$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-2015}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $2$ | \(\Q(\sqrt{403}) \) | \(\Z/4\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-5}) \) | \(\Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-5}, \sqrt{403})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | deg 8 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | deg 8 | \(\Z/8\Z\) | not in database |
| $8$ | deg 8 | \(\Z/8\Z\) | not in database |
| $8$ | deg 8 | \(\Z/6\Z\) | 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 | \(\Z/2\Z \oplus \Z/6\Z\) | 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 | add | nonsplit | nonsplit | ss | ord | nonsplit | ord | ord | ord | ord | split | ord | ord | ord | ord |
| $\lambda$-invariant(s) | - | 3 | 1 | 1,1 | 1 | 1 | 1 | 1 | 3 | 1 | 2 | 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
Note: $p$-adic regulator data only exists for primes $p\ge 5$ of good ordinary reduction.