Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-x^2-19821569x-33950770815\)
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(homogenize, simplify) |
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\(y^2z=x^3-x^2z-19821569xz^2-33950770815z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-1605547116x-24754928565456\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(-36146319240767/14232251401, 2620260993124267616/1697893359887899)$ | $28.243860616005257574691844752$ | $\infty$ |
| $(-2535, 0)$ | $0$ | $2$ |
Integral points
\( \left(-2535, 0\right) \)
Invariants
| Conductor: | $N$ | = | \( 25536 \) | = | $2^{6} \cdot 3 \cdot 7 \cdot 19$ |
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| Discriminant: | $\Delta$ | = | $277696187061542387712$ | = | $2^{16} \cdot 3^{7} \cdot 7^{10} \cdot 19^{3} $ |
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| j-invariant: | $j$ | = | \( \frac{13141891860831409148932}{4237307541832617} \) | = | $2^{2} \cdot 3^{-7} \cdot 7^{-10} \cdot 17^{3} \cdot 19^{-3} \cdot 149^{3} \cdot 5869^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.8971524445662166485195995878$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $1.9729562038196229026299567592$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0328439560266376$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $6.1116867547816645$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $28.243860616005257574691844752$ |
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| Real period: | $\Omega$ | ≈ | $0.071552275402707808689012820638$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 8 $ = $ 2^{2}\cdot1\cdot2\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $4.0418249864642016180391459581 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 4.041824986 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.071552 \cdot 28.243861 \cdot 8}{2^2} \\ & \approx 4.041824986\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 1505280 |
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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 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 | 6 | 16 | 0 |
| $3$ | $1$ | $I_{7}$ | nonsplit multiplicative | 1 | 1 | 7 | 7 |
| $7$ | $2$ | $I_{10}$ | nonsplit multiplicative | 1 | 1 | 10 | 10 |
| $19$ | $1$ | $I_{3}$ | nonsplit multiplicative | 1 | 1 | 3 | 3 |
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 | 2.3.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 3192 = 2^{3} \cdot 3 \cdot 7 \cdot 19 \), index $12$, genus $0$, and generators
$\left(\begin{array}{rr} 1597 & 4 \\ 2 & 9 \end{array}\right),\left(\begin{array}{rr} 401 & 2794 \\ 2792 & 399 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 2 & 5 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1066 & 1 \\ 1063 & 0 \end{array}\right),\left(\begin{array}{rr} 3189 & 4 \\ 3188 & 5 \end{array}\right),\left(\begin{array}{rr} 2018 & 1 \\ 2183 & 0 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 4 & 1 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 8 & 11 \end{array}\right),\left(\begin{array}{rr} 913 & 4 \\ 1826 & 9 \end{array}\right)$.
The torsion field $K:=\Q(E[3192])$ is a degree-$1525001748480$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/3192\Z)$.
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 | $2$ | \( 57 = 3 \cdot 19 \) |
| $3$ | nonsplit multiplicative | $4$ | \( 448 = 2^{6} \cdot 7 \) |
| $5$ | good | $2$ | \( 3648 = 2^{6} \cdot 3 \cdot 19 \) |
| $7$ | nonsplit multiplicative | $8$ | \( 1216 = 2^{6} \cdot 19 \) |
| $19$ | nonsplit multiplicative | $20$ | \( 1344 = 2^{6} \cdot 3 \cdot 7 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2.
Its isogeny class 25536.l
consists of 2 curves linked by isogenies of
degree 2.
Twists
The minimal quadratic twist of this elliptic curve is 3192.p1, its twist by $8$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{57}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $4$ | 4.0.178752.1 | \(\Z/4\Z\) | not in database |
| $8$ | 8.0.103812949610496.4 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | deg 8 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.2.27874423406592.23 | \(\Z/6\Z\) | 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 | add | nonsplit | ord | nonsplit | ord | ord | ss | nonsplit | ord | ord | ord | ord | ord | ord | ord |
| $\lambda$-invariant(s) | - | 3 | 1 | 1 | 1 | 1 | 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 | 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.