Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3+x^2-14404x+653827\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3+x^2z-14404xz^2+653827z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-18668259x+30784973022\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(74, -37\right) \) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([74:-37:1]\) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(2679, 0\right) \) | $0$ | $2$ |
Integral points
\( \left(74, -37\right) \)
\([74:-37:1]\)
\( \left(2679, 0\right) \)
Invariants
| Conductor: | $N$ | = | \( 72897 \) | = | $3 \cdot 11 \cdot 47^{2}$ |
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| Minimal Discriminant: | $\Delta$ | = | $3201426952713$ | = | $3^{3} \cdot 11 \cdot 47^{6} $ |
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| j-invariant: | $j$ | = | \( \frac{30664297}{297} \) | = | $3^{-3} \cdot 11^{-1} \cdot 313^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.2192945531783953857507970932$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.70577924767663390765967824169$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0970565641289056$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $3.6027691144463394$ | |||
| Intrinsic torsion order: | $\#E(\mathbb Q)_\text{tors}^\text{is}$ | = | $2$ | |||
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: | $\Omega$ | ≈ | $0.80069184270320086538511263327$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 2 $ = $ 1\cdot1\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L(E,1)$ | ≈ | $3.6031132921644038942330068497 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $9$ = $3^2$ (exact) |
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BSD formula
$$\begin{aligned} 3.603113292 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{9 \cdot 0.800692 \cdot 1.000000 \cdot 2}{2^2} \\ & \approx 3.603113292\end{aligned}$$
Modular invariants
Modular form 72897.2.a.a
For more coefficients, see the Downloads section to the right.
| Modular degree: | 154560 |
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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 3 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $3$ | $1$ | $I_{3}$ | nonsplit multiplicative | 1 | 1 | 3 | 3 |
| $11$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
| $47$ | $2$ | $I_0^{*}$ | 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 | $\ell$-adic index |
|---|---|---|---|
| $2$ | 2B | 4.6.0.1 | $6$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 12408 = 2^{3} \cdot 3 \cdot 11 \cdot 47 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 11087 & 0 \\ 0 & 12407 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 7 & 6 \\ 12402 & 12403 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 4 & 17 \end{array}\right),\left(\begin{array}{rr} 12401 & 8 \\ 12400 & 9 \end{array}\right),\left(\begin{array}{rr} 10388 & 8977 \\ 12079 & 7662 \end{array}\right),\left(\begin{array}{rr} 6440 & 2115 \\ 6533 & 8978 \end{array}\right),\left(\begin{array}{rr} 11657 & 564 \\ 6110 & 1223 \end{array}\right),\left(\begin{array}{rr} 3667 & 3666 \\ 9682 & 4795 \end{array}\right)$.
The torsion field $K:=\Q(E[12408])$ is a degree-$96787641139200$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/12408\Z)$.
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 |
|---|---|---|---|
| $3$ | nonsplit multiplicative | $4$ | \( 24299 = 11 \cdot 47^{2} \) |
| $11$ | nonsplit multiplicative | $12$ | \( 6627 = 3 \cdot 47^{2} \) |
| $47$ | additive | $1106$ | \( 33 = 3 \cdot 11 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2 and 4.
Its isogeny class 72897a
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
The minimal quadratic twist of this elliptic curve is 33a2, its twist by $-47$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{33}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $2$ | \(\Q(\sqrt{141}) \) | \(\Z/4\Z\) | not in database |
| $2$ | \(\Q(\sqrt{517}) \) | \(\Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{33}, \sqrt{141})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | deg 8 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.8.110190536020224.1 | \(\Z/8\Z\) | not in database |
| $8$ | 8.0.77801872968369.3 | \(\Z/8\Z\) | 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 | 11 | 47 |
|---|---|---|---|---|
| Reduction type | ord | nonsplit | nonsplit | add |
| $\lambda$-invariant(s) | 10 | 2 | 0 | - |
| $\mu$-invariant(s) | 0 | 0 | 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 $0$.