Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-x^2+7775x+5624641\)
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(homogenize, simplify) |
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\(y^2z=x^3-x^2z+7775xz^2+5624641z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3+629748x+4102252560\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(-35, 2304\right) \) | $1.7699289568482357412839756096$ | $\infty$ |
| \( \left(-163, 0\right) \) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([-35:2304:1]\) | $1.7699289568482357412839756096$ | $\infty$ |
| \([-163:0:1]\) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(-318, 62208\right) \) | $1.7699289568482357412839756096$ | $\infty$ |
| \( \left(-1470, 0\right) \) | $0$ | $2$ |
Integral points
\( \left(-163, 0\right) \), \((-35,\pm 2304)\), \((523,\pm 12348)\)
\([-163:0:1]\), \([-35:\pm 2304:1]\), \([523:\pm 12348:1]\)
\( \left(-163, 0\right) \), \((-35,\pm 2304)\), \((523,\pm 12348)\)
Invariants
| Conductor: | $N$ | = | \( 9408 \) | = | $2^{6} \cdot 3 \cdot 7^{2}$ |
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| Minimal Discriminant: | $\Delta$ | = | $-13709678915616768$ | = | $-1 \cdot 2^{22} \cdot 3^{4} \cdot 7^{9} $ |
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| j-invariant: | $j$ | = | \( \frac{4913}{1296} \) | = | $2^{-4} \cdot 3^{-4} \cdot 17^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.7755833047251611573793236903$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.72357007790624178557553904947$ |
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| $abc$ quality: | $Q$ | ≈ | $1.1327926724636421$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.8757091580653915$ | |||
| Intrinsic torsion order: | $\#E(\mathbb Q)_\text{tors}^\text{is}$ | = | $1$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $1.7699289568482357412839756096$ |
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| Real period: | $\Omega$ | ≈ | $0.30742016307851792720036808049$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 16 $ = $ 2^{2}\cdot2\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $2.1764473942067030033263152116 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 2.176447394 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.307420 \cdot 1.769929 \cdot 16}{2^2} \\ & \approx 2.176447394\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 86016 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $4$ | $I_{12}^{*}$ | additive | -1 | 6 | 22 | 4 |
| $3$ | $2$ | $I_{4}$ | nonsplit multiplicative | 1 | 1 | 4 | 4 |
| $7$ | $2$ | $III^{*}$ | additive | -1 | 2 | 9 | 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 | 8.48.0.203 | $48$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has label 56.96.1.co.2, level \( 56 = 2^{3} \cdot 7 \), index $96$, genus $1$, and generators
$\left(\begin{array}{rr} 3 & 4 \\ 52 & 51 \end{array}\right),\left(\begin{array}{rr} 49 & 8 \\ 48 & 9 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 28 & 29 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 5 & 4 \\ 52 & 53 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 15 \end{array}\right),\left(\begin{array}{rr} 24 & 5 \\ 27 & 32 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[56])$ is a degree-$32256$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/56\Z)$.
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 | $2$ | \( 7 \) |
| $3$ | nonsplit multiplicative | $4$ | \( 3136 = 2^{6} \cdot 7^{2} \) |
| $7$ | additive | $20$ | \( 192 = 2^{6} \cdot 3 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2.
Its isogeny class 9408ch
consists of 2 curves linked by isogenies of
degree 2.
Twists
The minimal quadratic twist of this elliptic curve is 294g1, its twist by $56$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-7}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $4$ | 4.2.87808.1 | \(\Z/8\Z\) | not in database |
| $4$ | 4.0.21952.1 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.7710244864.3 | \(\Z/4\Z \oplus \Z/8\Z\) | not in database |
| $8$ | 8.2.341461686730752.19 | \(\Z/6\Z\) | 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 |
| $16$ | 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 | add | ord | ord | ss | ord | ss | ord | ord | ord | ss | ord | ord |
| $\lambda$-invariant(s) | - | 5 | 1 | - | 1 | 1 | 1,3 | 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 | 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.