Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-19058403x+31172624098\)
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(homogenize, simplify) |
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\(y^2z=x^3-19058403xz^2+31172624098z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-19058403x+31172624098\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{2}\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(2177, 0)$ | $0$ | $2$ |
| $(2849, 0)$ | $0$ | $2$ |
Integral points
\( \left(-5026, 0\right) \), \( \left(2177, 0\right) \), \( \left(2849, 0\right) \)
Invariants
| Conductor: | $N$ | = | \( 35280 \) | = | $2^{4} \cdot 3^{2} \cdot 5 \cdot 7^{2}$ |
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| Discriminant: | $\Delta$ | = | $23248047443394816000000$ | = | $2^{14} \cdot 3^{8} \cdot 5^{6} \cdot 7^{12} $ |
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| j-invariant: | $j$ | = | \( \frac{2179252305146449}{66177562500} \) | = | $2^{-2} \cdot 3^{-2} \cdot 5^{-6} \cdot 7^{-6} \cdot 13^{3} \cdot 9973^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $3.0678608221776089117848576516$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.85245242275595210411732653996$ |
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| $abc$ quality: | $Q$ | ≈ | $1.016290927006418$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $5.911778843914741$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 0$ |
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| Mordell-Weil rank: | $r$ | = | $ 0$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | = | $1$ |
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| Real period: | $\Omega$ | ≈ | $0.11958178973959386917448650121$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 128 $ = $ 2^{2}\cdot2^{2}\cdot2\cdot2^{2} $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $4$ |
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| Special value: | $ L(E,1)$ | ≈ | $0.95665431791675095339589200971 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $1$ (exact) |
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BSD formula
$$\begin{aligned} 0.956654318 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.119582 \cdot 1.000000 \cdot 128}{4^2} \\ & \approx 0.956654318\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 2654208 |
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| $ \Gamma_0(N) $-optimal: | no | |
| Manin constant: | 1 |
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Local data at primes of bad reduction
This elliptic curve is not semistable. There are 4 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $4$ | $I_{6}^{*}$ | additive | -1 | 4 | 14 | 2 |
| $3$ | $4$ | $I_{2}^{*}$ | additive | -1 | 2 | 8 | 2 |
| $5$ | $2$ | $I_{6}$ | nonsplit multiplicative | 1 | 1 | 6 | 6 |
| $7$ | $4$ | $I_{6}^{*}$ | additive | -1 | 2 | 12 | 6 |
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 | 2.6.0.1 |
| $3$ | 3B | 3.4.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 840 = 2^{3} \cdot 3 \cdot 5 \cdot 7 \), index $384$, genus $5$, and generators
$\left(\begin{array}{rr} 337 & 6 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 12 & 1 \end{array}\right),\left(\begin{array}{rr} 599 & 828 \\ 234 & 767 \end{array}\right),\left(\begin{array}{rr} 829 & 12 \\ 828 & 13 \end{array}\right),\left(\begin{array}{rr} 9 & 4 \\ 824 & 833 \end{array}\right),\left(\begin{array}{rr} 623 & 834 \\ 48 & 41 \end{array}\right),\left(\begin{array}{rr} 1 & 12 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 427 & 6 \\ 834 & 835 \end{array}\right),\left(\begin{array}{rr} 559 & 834 \\ 140 & 839 \end{array}\right)$.
The torsion field $K:=\Q(E[840])$ is a degree-$185794560$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/840\Z)$.
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 | $2$ | \( 441 = 3^{2} \cdot 7^{2} \) |
| $3$ | additive | $8$ | \( 784 = 2^{4} \cdot 7^{2} \) |
| $5$ | nonsplit multiplicative | $6$ | \( 7056 = 2^{4} \cdot 3^{2} \cdot 7^{2} \) |
| $7$ | additive | $32$ | \( 720 = 2^{4} \cdot 3^{2} \cdot 5 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 3 and 6.
Its isogeny class 35280dy
consists of 8 curves linked by isogenies of
degrees dividing 12.
Twists
The minimal quadratic twist of this elliptic curve is 210a6, its twist by $-84$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{7}) \) | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-3}, \sqrt{-35})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{3}, \sqrt{-14})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{30}, \sqrt{35})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $6$ | 6.0.15554923776.3 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $8$ | 8.8.7965941760000.9 | \(\Z/2\Z \oplus \Z/12\Z\) | not in database |
| $8$ | 8.0.31116960000.2 | \(\Z/2\Z \oplus \Z/12\Z\) | not in database |
| $8$ | 8.0.12745506816.3 | \(\Z/2\Z \oplus \Z/12\Z\) | not in database |
| $12$ | deg 12 | \(\Z/6\Z \oplus \Z/6\Z\) | not in database |
| $16$ | 16.0.63456228123711897600000000.4 | \(\Z/4\Z \oplus \Z/4\Z\) | 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 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $18$ | 18.6.1000940353506452966226529287168000000000000.3 | \(\Z/2\Z \oplus \Z/18\Z\) | 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 |
|---|---|---|---|---|
| Reduction type | add | add | nonsplit | add |
| $\lambda$-invariant(s) | - | - | 0 | - |
| $\mu$-invariant(s) | - | - | 0 | - |
All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 5$ of good reduction are zero.
An entry - indicates that the invariants are not computed because the reduction is additive.
$p$-adic regulators
All $p$-adic regulators are identically $1$ since the rank is $0$.